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Ingy döt Net 2026-02-01 16:33:20 -08:00
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1
Task/Miller-Rabin-primality-test/00-META.yaml
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---
from: http://rosettacode.org/wiki/Miller–Rabin_primality_test
note: Prime Numbers

25
Task/Miller-Rabin-primality-test/00-TASK.txt
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The [[wp:Miller–Rabin primality test|Miller–Rabin primality test]] or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime or not.
The algorithm, as modified by [[wp:Michael O. Rabin|Michael O. Rabin]] to avoid the [[wp:generalized Riemann hypothesis|generalized Riemann hypothesis]], is a probabilistic algorithm.
The pseudocode, from [[wp:Miller-Rabin primality test#Algorithm_and_running_time|Wikipedia]] is:
'''Input''': ''n'' > 2, an odd integer to be tested for primality;
''k'', a parameter that determines the accuracy of the test
'''Output''': ''composite'' if ''n'' is composite, otherwise ''probably prime''
write ''n'' − 1 as 2<sup>''s''</sup>·''d'' with ''d'' odd by factoring powers of 2 from ''n'' − 1
LOOP: '''repeat''' ''k'' times:
pick ''a'' randomly in the range [2, ''n'' − 1]
''x'' ← ''a''<sup>''d''</sup> mod ''n''
'''if''' ''x'' = 1 or ''x'' = ''n'' − 1 '''then''' '''do''' '''next''' LOOP
'''repeat''' ''s'' − 1 times:
''x'' ← ''x''<sup>2</sup> mod ''n''
'''if''' ''x'' = 1 '''then''' '''return''' ''composite''
'''if''' ''x'' = ''n'' − 1 '''then''' '''do''' '''next''' LOOP
'''return''' ''composite''
'''return''' ''probably prime''
* The nature of the test involves big numbers, so the use of "big numbers" libraries (or similar features of the language of your choice) are suggested, but '''not''' mandatory.
* Deterministic variants of the test exist and can be implemented as extra (not mandatory to complete the task)
<br><br>
#REDIRECT [[Miller-Rabin primality test]]

29
Task/Miller-Rabin-primality-test/11l/miller-rabin-primality-test.11l
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F isProbablePrime(n, k = 10)
I n < 2 | n % 2 == 0
R n == 2
V d = n - 1
V s = 0
L d % 2 == 0
d I/= 2
s++
assert(2 ^ s * d == n - 1)
L 0 .< k
V a = random:(2 .< n)
V x = pow(a, d, n)
I x == 1 | x == n - 1
L.continue
L 0 .< s - 1
x = pow(x, 2, n)
I x == 1
R 0B
I x == n - 1
L.break
L.was_no_break
R 0B
R 1B
print((2..29).filter(x -> isProbablePrime(x)))

298
Task/Miller-Rabin-primality-test/AArch64-Assembly/miller-rabin-primality-test.aarch64
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/* ARM assembly AARCH64 Raspberry PI 3B */
/* program testmiller64B.s */
// optimisation : one routine
/************************************/
/* Constantes */
/************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
.equ NBLOOP, 5 // loop number change this if necessary
// if modify, thinck to add test to little prime value
// in routine
//.include "../../ficmacros64.inc" // use for developper debugging
/*******************************************/
/* Initialized data */
/*******************************************/
.data
szMessStartPgm: .asciz "Program 64 bits start \n"
szMessEndPgm: .asciz "Program normal end.\n"
szMessPrime: .asciz " is prime !!!.\n"
szMessNotPrime: .asciz " is not prime !!!.\n"
szCarriageReturn: .asciz "\n"
/*******************************************/
/* UnInitialized data */
/*******************************************/
.bss
.align 4
sZoneConv: .skip 24
/*******************************************/
/* code section */
/*******************************************/
.text
.global main
main: // program start
ldr x0,qAdrszMessStartPgm // display start message
bl affichageMess
ldr x4,iStart // start number
ldr x5,iLimit // end number
tst x4,#1
cinc x4,x4,eq // start with odd number
1:
mov x0,x4
bl isPrimeMiller // test miller rabin
cmp x0,#0
beq 2f
mov x0,x4
ldr x1,qAdrsZoneConv
bl conversion10 // decimal conversion
ldr x0,qAdrsZoneConv
bl affichageMess
ldr x0,qAdrszMessPrime
bl affichageMess
b 3f
2:
ldr x0,qAdrszMessNotPrime
// bl affichageMess
3:
add x4,x4,#2
cmp x4,x5
blo 1b
ldr x0,qAdrszMessEndPgm // display end message
bl affichageMess
100: // standard end of the program
mov x0, #0 // return code
mov x8, #EXIT // request to exit program
svc 0 // perform system call
qAdrszMessStartPgm: .quad szMessStartPgm
qAdrszMessEndPgm: .quad szMessEndPgm
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrsZoneConv: .quad sZoneConv
qAdrszMessPrime: .quad szMessPrime
qAdrszMessNotPrime: .quad szMessNotPrime
//iStart: .quad 0x0
//iLimit: .quad 0x100
iStart: .quad 0xFFFFFFFFFFFFFF00
iLimit: .quad 0xFFFFFFFFFFFFFFF0
//iStart: .quad 341550071728360
//iLimit: .quad 341550071728380
//359341550071728361
/***************************************************/
/* test miller rabin algorithme wikipedia */
/* unsigned */
/***************************************************/
/* x0 contains number */
/* x1 contains parameter */
/* x0 return 1 if prime 0 if composite */
isPrimeMiller:
stp x1,lr,[sp,-16]! // TODO: save à completer
stp x2,x3,[sp,-16]!
stp x4,x5,[sp,-16]!
stp x6,x7,[sp,-16]!
stp x8,x9,[sp,-16]!
cmp x0,#1 // control 0 or 1
csel x0,xzr,x0,ls
bls 100f
cmp x0,#2 // control = 2
mov x1,1
csel x0,x1,x0,eq
beq 100f
cmp x0,#3 // control = 3
csel x0,x1,x0,eq
beq 100f
cmp x0,#5 // control = 5
csel x0,x1,x0,eq
beq 100f
cmp x0,#7 // control = 7
csel x0,x1,x0,eq
beq 100f
cmp x0,#11 // control = 11
csel x0,x1,x0,eq
beq 100f
tst x0,#1
csel x0,xzr,x0,eq // even
beq 100f
mov x4,x0 // N
sub x3,x0,#1 // D
mov x2,#2
mov x6,#0 // S
1: // compute D * 2 power S
lsr x3,x3,#1 // D= D/2
add x6,x6,#1 // increment S
tst x3,#1 // D even ?
beq 1b
2:
mov x8,#0 // loop counter
sub x5,x0,#3
mov x7,3
3:
mov x0,x7
mov x1,x3 // exposant = D
mov x2,x4 // modulo N
bl moduloPur64
cmp x0,#1
beq 5f
sub x1,x4,#1 // n -1
cmp x0,x1
beq 5f
sub x9,x6,#1 // S - 1
4:
mov x2,x0
mul x0,x2,x2 // compute square lower
umulh x1,x2,x2 // compute square upper
mov x2,x4 // and compute modulo N
bl division64R2023
mov x0,x2
cmp x0,#1
csel x0,xzr,x0,eq // composite
beq 100f
sub x1,x4,#1 // n -1
cmp x0,x1
beq 5f
subs x9,x9,#1
bge 4b
mov x0,#0 // composite
b 100f
5:
add x7,x7,2
add x8,x8,#1
cmp x8,NBLOOP
blt 3b
mov x0,#1 // prime
100:
ldp x8,x9,[sp],16
ldp x6,x7,[sp],16
ldp x4,x5,[sp],16
ldp x2,x3,[sp],16
ldp x1,lr,[sp],16 // TODO: retaur à completer
ret
/********************************************************/
/* Calcul modulo de b puissance e modulo m */
/* Exemple 4 puissance 13 modulo 497 = 445 */
/********************************************************/
/* x0 nombre */
/* x1 exposant */
/* x2 modulo */
moduloPur64:
stp x1,lr,[sp,-16]! // save registres
stp x3,x4,[sp,-16]! // save registres
stp x5,x6,[sp,-16]! // save registres
stp x7,x8,[sp,-16]! // save registres
stp x9,x10,[sp,-16]! // save registres
cbz x0,100f
cbz x1,100f
mov x8,x0
mov x7,x1
mov x10,x2 // modulo
mov x6,1 // resultat
udiv x4,x8,x10
msub x9,x4,x10,x8 // contient le reste
1:
tst x7,1
beq 2f
mul x4,x9,x6
umulh x5,x9,x6
mov x6,x4
mov x0,x6
mov x1,x5
mov x2,x10
bl division64R2023
mov x6,x2
2:
mul x8,x9,x9
umulh x5,x9,x9
mov x0,x8
mov x1,x5
mov x2,x10
bl division64R2023
mov x9,x2
lsr x7,x7,1
cbnz x7,1b
mov x0,x6 // result
cmn x0,0 // clear carry not error
100:
ldp x9,x10,[sp],16 // restaur des 2 registres
ldp x7,x8,[sp],16 // restaur des 2 registres
ldp x5,x6,[sp],16 // restaur des 2 registres
ldp x3,x4,[sp],16 // restaur des 2 registres
ldp x1,lr,[sp],16 // restaur des 2 registres
ret // retour adresse lr x30
/***************************************************/
/* division number 128 bits in 2 registers by number 64 bits */
/* unsigned */
/***************************************************/
/* x0 contains lower part dividende */
/* x1 contains upper part dividende */
/* x2 contains divisor */
/* x0 return lower part quotient */
/* x1 return upper part quotient */
/* x2 return remainder */
division64R2023:
stp x3,lr,[sp,-16]!
stp x4,x5,[sp,-16]!
stp x6,x7,[sp,-16]!
mov x4,x2 // save divisor
mov x5,#0 // init upper part divisor
mov x2,x0 // save dividende
mov x3,x1
mov x0,#0 // init result
mov x1,#0
mov x6,#0 // init shift counter
1: // loop shift divisor
cmp x5,#0 // upper divisor <0
blt 2f
cmp x5,x3
bhi 2f
blo 11f
cmp x4,x2
bhi 2f // new divisor > dividende
11:
lsl x5,x5,#1 // shift left one bit upper divisor
tst x4,#0x8000000000000000
lsl x4,x4,#1 // shift left one bit lower divisor
orr x7,x5,#1
csel x5,x7,x5,ne // move bit 63 lower on upper
add x6,x6,#1 // increment shift counter
b 1b
2: // loop 2
lsl x1,x1,#1 // shift left one bit upper quotient
tst x0,#0x8000000000000000
lsl x0,x0,#1 // shift left one bit lower quotient
orr x7,x1,#1
csel x1,x7,x1,ne // move bit 63 lower on upper
cmp x5,x3 // compare divisor and dividende
bhi 3f
blo 21f
cmp x4,x2
bhi 3f
21:
subs x2,x2,x4 // < sub divisor from dividende lower
sbc x3,x3,x5 // and upper
orr x0,x0,#1 // move 1 on quotient
3:
lsr x4,x4,#1 // shift right one bit upper divisor
tst x5,1
lsr x5,x5,#1 // and lower
orr x7,x4,#0x8000000000000000 // move bit 0 upper to 31 bit lower
csel x4,x7,x4,ne // move bit 0 upper to 63 bit lower
subs x6,x6,#1 // decrement shift counter
bge 2b // if > 0 loop 2
100:
ldp x6,x7,[sp],16
ldp x4,x5,[sp],16
ldp x3,lr,[sp],16
ret
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
.include "../includeARM64.inc"

38
Task/Miller-Rabin-primality-test/ALGOL-68/miller-rabin-primality-test.alg
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MODE LINT=LONG INT;
MODE LOOPINT = INT;
MODE POWMODSTRUCT = LINT;
PR READ "prelude/pow_mod.a68" PR;
PROC miller rabin = (LINT n, LOOPINT k)BOOL: (
IF n<=3 THEN TRUE
ELIF NOT ODD n THEN FALSE
ELSE
LINT d := n - 1;
INT s := 0;
WHILE NOT ODD d DO
d := d OVER 2;
s +:= 1
OD;
TO k DO
LINT a := 2 + ENTIER (random*(n-3));
LINT x := pow mod(a, d, n);
IF x /= 1 THEN
TO s DO
IF x = n-1 THEN done FI;
x := x*x %* n
OD;
else: IF x /= n-1 THEN return false FI;
done: EMPTY
FI
OD;
TRUE EXIT
return false: FALSE
FI
);
FOR i FROM 937 TO 1000 DO
IF miller rabin(i, 10) THEN
print((" ",whole(i,0)))
FI
OD

286
Task/Miller-Rabin-primality-test/ARM-Assembly/miller-rabin-primality-test.arm
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/* ARM assembly Raspberry PI */
/* program testmiller.s */
/* for constantes see task include a file in arm assembly */
/************************************/
/* Constantes */
/************************************/
.include "../constantes.inc"
.equ NBDIVISORS, 2000
//.include "../../ficmacros32.inc" @ use for developper debugging
/*******************************************/
/* Initialized data */
/*******************************************/
.data
szMessStartPgm: .asciz "Program 32 bits start \n"
szMessEndPgm: .asciz "Program normal end.\n"
szMessErrorArea: .asciz "\033[31mError : area divisors too small.\n"
szMessPrime: .asciz " is prime !!!.\n"
szMessNotPrime: .asciz " is not prime !!!.\n"
szCarriageReturn: .asciz "\n"
.align 4
iGraine: .int 123456
/*******************************************/
/* UnInitialized data */
/*******************************************/
.bss
.align 4
sZoneConv: .skip 24
/*******************************************/
/* code section */
/*******************************************/
.text
.global main
main: @ program start
ldr r0,iAdrszMessStartPgm @ display start message
bl affichageMess
ldr r4,iStart @ start number
ldr r5,iLimit @ end number
tst r4,#1
addeq r4,#1 @ start with odd number
1:
mov r0,r4
ldr r1,iAdrsZoneConv
bl conversion10 @ decimal conversion
ldr r0,iAdrsZoneConv
bl affichageMess
mov r0,r4
bl isPrimeMiller @ test miller rabin
cmp r0,#0
beq 2f
ldr r0,iAdrszMessPrime
bl affichageMess
b 3f
2:
ldr r0,iAdrszMessNotPrime
bl affichageMess
3:
add r4,r4,#2
cmp r4,r5
ble 1b
ldr r0,iAdrszMessEndPgm @ display end message
bl affichageMess
100: @ standard end of the program
mov r0, #0 @ return code
mov r7, #EXIT @ request to exit program
svc 0 @ perform system call
iAdrszMessStartPgm: .int szMessStartPgm
iAdrszMessEndPgm: .int szMessEndPgm
iAdrszCarriageReturn: .int szCarriageReturn
iAdrsZoneConv: .int sZoneConv
iAdrszMessPrime: .int szMessPrime
iAdrszMessNotPrime: .int szMessNotPrime
iStart: .int 4294967270
iLimit: .int 4294967295
/***************************************************/
/* check if a number is prime test miller rabin */
/***************************************************/
/* r0 contains the number */
/* r0 return 1 if prime 0 else */
@2147483647
@4294967297
@131071
isPrimeMiller:
push {r1-r6,lr} @ save registers
cmp r0,#1 @ control 0 or 1
movls r0,#0
bls 100f
cmp r0,#2 @ control = 2
moveq r0,#1
beq 100f
tst r0,#1
moveq r0,#0 @ even
beq 100f
mov r1,#5 @ loop number
bl testMiller
100:
pop {r1-r6,pc}
/***************************************************/
/* test miller rabin algorithme wikipedia */
/* unsigned */
/***************************************************/
/* r0 contains number */
/* r1 contains parameter */
/* r0 return 1 if prime 0 if composite */
testMiller:
push {r1-r9,lr} @ save registers
mov r4,r0 @ N
mov r7,r1 @ loop maxi
sub r3,r0,#1 @ D
mov r2,#2
mov r6,#0 @ S
1: @ compute D * 2 power S
lsr r3,#1 @ D= D/2
add r6,r6,#1 @ increment S
tst r3,#1 @ D even ?
beq 1b
2:
mov r8,#0 @ loop counter
sub r5,r0,#3
3:
mov r0,r5
bl genereraleas
add r0,r0,#2 @ alea (entre 2 et n-1)
mov r1,r3 @ exposant = D
mov r2,r4 @ modulo N
bl moduloPuR32
cmp r0,#1
beq 5f
sub r1,r4,#1 @ n -1
cmp r0,r1
beq 5f
sub r9,r6,#1 @ S - 1
4:
mov r2,r0
umull r0,r1,r2,r0 @ compute square
mov r2,r4 @ and compute modulo N
bl division32R2023
mov r0,r2
cmp r0,#1
moveq r0,#0 @ composite
beq 100f
sub r1,r4,#1 @ n -1
cmp r0,r1
beq 5f
subs r9,r9,#1
bge 4b
mov r0,#0 @ composite
b 100f
5:
add r8,r8,#1
cmp r8,r7
blt 3b
mov r0,#1 @ prime
100:
pop {r1-r9,pc}
/********************************************************/
/* Calcul modulo de b puissance e modulo m */
/* Exemple 4 puissance 13 modulo 497 = 445 */
/* */
/********************************************************/
/* r0 nombre */
/* r1 exposant */
/* r2 modulo */
/* r0 return result */
moduloPuR32:
push {r1-r6,lr} @ save registers
cmp r0,#0 @ control <> zero
beq 100f
cmp r2,#0 @ control <> zero
beq 100f
1:
mov r4,r2 @ save modulo
mov r5,r1 @ save exposant
mov r6,r0 @ save base
mov r3,#1 @ start result
mov r1,#0 @ division r0,r1 by r2
bl division32R2023
mov r6,r2 @ base <- remainder
2:
tst r5,#1 @ exposant even or odd
beq 3f
umull r0,r1,r6,r3 @ multiplication base
mov r2,r4 @ and compute modulo N
bl division32R2023
mov r3,r2 @ result <- remainder
3:
umull r0,r1,r6,r6 @ compute base square
mov r2,r4 @ and compute modulo N
bl division32R2023
mov r6,r2 @ base <- remainder
lsr r5,#1 @ left shift 1 bit
cmp r5,#0 @ end ?
bne 2b
mov r0,r3
100:
pop {r1-r6,pc}
/***************************************************/
/* division number 64 bits in 2 registers by number 32 bits */
/* unsigned */
/***************************************************/
/* r0 contains lower part dividende */
/* r1 contains upper part dividende */
/* r2 contains divisor */
/* r0 return lower part quotient */
/* r1 return upper part quotient */
/* r2 return remainder */
division32R2023:
push {r3-r6,lr} @ save registers
mov r4,r2 @ save divisor
mov r5,#0 @ init upper part divisor
mov r2,r0 @ save dividende
mov r3,r1
mov r0,#0 @ init result
mov r1,#0
mov r6,#0 @ init shift counter
1: @ loop shift divisor
cmp r5,#0 @ upper divisor <0
blt 2f
cmp r5,r3
cmpeq r4,r2
bhs 2f @ new divisor > dividende
lsl r5,#1 @ shift left one bit upper divisor
lsls r4,#1 @ shift left one bit lower divisor
orrcs r5,r5,#1 @ move bit 31 lower on upper
add r6,r6,#1 @ increment shift counter
b 1b
2: @ loop 2
lsl r1,#1 @ shift left one bit upper quotient
lsls r0,#1 @ shift left one bit lower quotient
orrcs r1,#1 @ move bit 31 lower on upper
cmp r5,r3 @ compare divisor and dividende
cmpeq r4,r2
bhi 3f
subs r2,r2,r4 @ < sub divisor from dividende lower
sbc r3,r3,r5 @ and upper
orr r0,r0,#1 @ move 1 on quotient
3:
lsr r4,r4,#1 @ shift right one bit upper divisor
lsrs r5,#1 @ and lower
orrcs r4,#0x80000000 @ move bit 0 upper to 31 bit lower
subs r6,#1 @ decrement shift counter
bge 2b @ if > 0 loop 2
100:
pop {r3-r6,pc}
/***************************************************/
/* Generation random number */
/***************************************************/
/* r0 contains limit */
genereraleas:
push {r1-r4,lr} @ save registers
ldr r4,iAdriGraine
ldr r2,[r4]
ldr r3,iNbDep1
mul r2,r3,r2
ldr r3,iNbDep1
add r2,r2,r3
str r2,[r4] @ save seed for next call
cmp r0,#0
beq 100f
mov r1,r0 @ divisor
mov r0,r2 @ dividende
bl division
mov r0,r3 @ résult = remainder
100: @ end function
pop {r1-r4,pc} @ restaur registers
iAdriGraine: .int iGraine
iNbDep1: .int 0x343FD
iNbDep2: .int 0x269EC3
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
.include "../affichage.inc"

9
Task/Miller-Rabin-primality-test/Ada/miller-rabin-primality-test-1.ada
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@ -1,9 +0,0 @@
generic
type Number is range <>;
package Miller_Rabin is
type Result_Type is (Composite, Probably_Prime);
function Is_Prime (N : Number; K : Positive := 10) return Result_Type;
end Miller_Rabin;

55
Task/Miller-Rabin-primality-test/Ada/miller-rabin-primality-test-2.ada
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with Ada.Numerics.Discrete_Random;
package body Miller_Rabin is
function Is_Prime (N : Number; K : Positive := 10)
return Result_Type
is
subtype Number_Range is Number range 2 .. N - 1;
package Random is new Ada.Numerics.Discrete_Random (Number_Range);
function Mod_Exp (Base, Exponent, Modulus : Number) return Number is
Result : Number := 1;
begin
for E in 1 .. Exponent loop
Result := Result * Base mod Modulus;
end loop;
return Result;
end Mod_Exp;
Generator : Random.Generator;
D : Number := N - 1;
S : Natural := 0;
X : Number;
begin
-- exclude 2 and even numbers
if N = 2 then
return Probably_Prime;
elsif N mod 2 = 0 then
return Composite;
end if;
-- write N-1 as 2**S * D, with D mod 2 /= 0
while D mod 2 = 0 loop
D := D / 2;
S := S + 1;
end loop;
-- initialize RNG
Random.Reset (Generator);
for Loops in 1 .. K loop
X := Mod_Exp(Random.Random (Generator), D, N);
if X /= 1 and X /= N - 1 then
Inner : for R in 1 .. S - 1 loop
X := Mod_Exp (X, 2, N);
if X = 1 then return Composite; end if;
exit Inner when X = N - 1;
end loop Inner;
if X /= N - 1 then return Composite; end if;
end if;
end loop;
return Probably_Prime;
end Is_Prime;
end Miller_Rabin;

25
Task/Miller-Rabin-primality-test/Ada/miller-rabin-primality-test-3.ada
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with Ada.Text_IO, Miller_Rabin;
procedure Mr_Tst is
type Number is range 0 .. (2**48)-1;
package Num_IO is new Ada.Text_IO.Integer_IO (Number);
package Pos_IO is new Ada.Text_IO.Integer_IO (Positive);
package MR is new Miller_Rabin(Number); use MR;
N : Number;
K : Positive;
begin
for I in Number(2) .. 1000 loop
if Is_Prime (I) = Probably_Prime then
Ada.Text_IO.Put (Number'Image (I));
end if;
end loop;
Ada.Text_IO.Put_Line (".");
Ada.Text_IO.Put ("Enter a Number: "); Num_IO.Get (N);
Ada.Text_IO.Put ("Enter the count of loops: "); Pos_IO.Get (K);
Ada.Text_IO.Put_Line ("What is it? " & Result_Type'Image (Is_Prime(N, K)));
end MR_Tst;

86
Task/Miller-Rabin-primality-test/Ada/miller-rabin-primality-test-4.ada
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with Ada.Text_IO, Crypto.Types.Big_Numbers, Ada.Numerics.Discrete_Random;
procedure Miller_Rabin is
Bound: constant Positive := 256; -- can be any multiple of 32
package LN is new Crypto.Types.Big_Numbers (Bound);
use type LN.Big_Unsigned; -- all computations "mod 2**Bound"
function "+"(S: String) return LN.Big_Unsigned
renames LN.Utils.To_Big_Unsigned;
function Is_Prime (N : LN.Big_Unsigned; K : Positive := 10) return Boolean is
subtype Mod_32 is Crypto.Types.Mod_Type;
use type Mod_32;
package R_32 is new Ada.Numerics.Discrete_Random (Mod_32);
Generator : R_32.Generator;
function Random return LN.Big_Unsigned is
X: LN.Big_Unsigned := LN.Big_Unsigned_Zero;
begin
for I in 1 .. Bound/32 loop
X := (X * 2**16) * 2**16;
X := X + R_32.Random(Generator);
end loop;
return X;
end Random;
D: LN.Big_Unsigned := N - LN.Big_Unsigned_One;
S: Natural := 0;
A, X: LN.Big_Unsigned;
begin
-- exclude 2 and even numbers
if N = 2 then
return True;
elsif N mod 2 = LN.Big_Unsigned_Zero then
return False;
else
-- write N-1 as 2**S * D, with odd D
while D mod 2 = LN.Big_Unsigned_Zero loop
D := D / 2;
S := S + 1;
end loop;
-- initialize RNG
R_32.Reset (Generator);
-- run the real test
for Loops in 1 .. K loop
loop
A := Random;
exit when (A > 1) and (A < (N - 1));
end loop;
X := LN.Mod_Utils.Pow(A, D, N); -- X := (Random**D) mod N
if X /= 1 and X /= N - 1 then
Inner:
for R in 1 .. S - 1 loop
X := LN.Mod_Utils.Pow(X, LN.Big_Unsigned_Two, N);
if X = 1 then
return False;
end if;
exit Inner when X = N - 1;
end loop Inner;
if X /= N - 1 then
return False;
end if;
end if;
end loop;
end if;
return True;
end Is_Prime;
S: constant String :=
"4547337172376300111955330758342147474062293202868155909489";
T: constant String :=
"4547337172376300111955330758342147474062293202868155909393";
K: constant Positive := 10;
begin
Ada.Text_IO.Put_Line("Prime(" & S & ")=" & Boolean'Image(Is_Prime(+S, K)));
Ada.Text_IO.Put_Line("Prime(" & T & ")=" & Boolean'Image(Is_Prime(+T, K)));
end Miller_Rabin;

26
Task/Miller-Rabin-primality-test/Ada/miller-rabin-primality-test-5.ada
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@ -1,26 +0,0 @@
with Ada.Text_IO, Crypto.Types.Big_Numbers, Ada.Numerics.Discrete_Random;
procedure Miller_Rabin is
Bound: constant Positive := 256; -- can be any multiple of 32
package LN is new Crypto.Types.Big_Numbers (Bound);
use type LN.Big_Unsigned; -- all computations "mod 2**Bound"
function "+"(S: String) return LN.Big_Unsigned
renames LN.Utils.To_Big_Unsigned;
S: constant String :=
"4547337172376300111955330758342147474062293202868155909489";
T: constant String :=
"4547337172376300111955330758342147474062293202868155909393";
K: constant Positive := 10;
begin
Ada.Text_IO.Put_Line("Prime(" & S & ")="
& Boolean'Image (LN.Mod_Utils.Passed_Miller_Rabin_Test(+S, K)));
Ada.Text_IO.Put_Line("Prime(" & T & ")="
& Boolean'Image (LN.Mod_Utils.Passed_Miller_Rabin_Test(+T, K)));
end Miller_Rabin;

41
Task/Miller-Rabin-primality-test/AutoHotkey/miller-rabin-primality-test.ahk
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@ -1,41 +0,0 @@
MsgBox % MillerRabin(999983,10) ; 1
MsgBox % MillerRabin(999809,10) ; 1
MsgBox % MillerRabin(999727,10) ; 1
MsgBox % MillerRabin(52633,10) ; 0
MsgBox % MillerRabin(60787,10) ; 0
MsgBox % MillerRabin(999999,10) ; 0
MsgBox % MillerRabin(999995,10) ; 0
MsgBox % MillerRabin(999991,10) ; 0
MillerRabin(n,k) { ; 0: composite, 1: probable prime (n < 2**31)
d := n-1, s := 0
While !(d&1)
d>>=1, s++
Loop %k% {
Random a, 2, n-2 ; if n < 4,759,123,141, it is enough to test a = 2, 7, and 61.
x := PowMod(a,d,n)
If (x=1 || x=n-1)
Continue
Cont := 0
Loop % s-1 {
x := PowMod(x,2,n)
If (x = 1)
Return 0
If (x = n-1) {
Cont = 1
Break
}
}
IfEqual Cont,1, Continue
Return 0
}
Return 1
}
PowMod(x,n,m) { ; x**n mod m
y := 1, i := n, z := x
While i>0
y := i&1 ? mod(y*z,m) : y, z := mod(z*z,m), i >>= 1
Return y
}

20
Task/Miller-Rabin-primality-test/BQN/miller-rabin-primality-test-1.bqn
View file

@ -1,20 +0,0 @@
_modMul ← { n _𝕣: n|× }
MillerRabin ← { 𝕊n: 10𝕊n ; iter 𝕊 n: !2|n
# n = 1 + d×2⋆s
s ← 0 {𝕨 2⊸|◶⟨+⟜1𝕊2⌊∘÷˜⊢,⊣⟩ 𝕩} n-1
d ← (n-1) ÷ 2⋆s
# Arithmetic mod n
Mul ← n _modMul
Pow ← Mul{𝔽´𝔽˜⍟(/2|⌊∘÷⟜2⍟(↕1+·⌊2⋆⁼⊢)𝕩)𝕨}
# Miller-Rabin test
MR ← {
1 =𝕩 ? 𝕨≠s ;
(n-1)=𝕩 ? 0 ;
𝕨≤1 ? 1 ;
(𝕨-1) 𝕊 Mul˜𝕩
}
C ← { 𝕊a: s MR a Pow d } # Is composite
{0:1; C •rand.Range⌾(-⟜2) n ? 0; 𝕊𝕩-1} iter
}

10
Task/Miller-Rabin-primality-test/BQN/miller-rabin-primality-test-2.bqn
View file

@ -1,10 +0,0 @@
# Compute n|𝕨×𝕩 in high precision
_modMul ← { n _𝕣:
# Split each argument into two 26-bit numbers, with the remaining
# mantissa bit encoded in the sign of the lower-order part.
q←1+2⋆27
Split ← { h←(q×𝕩)(⊣--)𝕩 ⋄ ⟨𝕩-h,h⟩ }
# The product, and an error relative to precise split multiplication.
Mul ← × (⊣ ⋈ -⊸(+´)) ·⥊×⌜○Split
((n×<⟜0)⊸+ -⟜n+⊢)´ n | Mul
}

4
Task/Miller-Rabin-primality-test/BQN/miller-rabin-primality-test-3.bqn
View file

@ -1,4 +0,0 @@
MillerRabin 15485867
1
MillerRabin¨⊸/ 101+2×↕10
⟨ 101 103 107 109 113 ⟩

68
Task/Miller-Rabin-primality-test/Bc/miller-rabin-primality-test.bc
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@ -1,68 +0,0 @@
seed = 1 /* seed of the random number generator */
scale = 0
/* Random number from 0 to 32767. */
define rand() {
/* Cheap formula (from POSIX) for random numbers of low quality. */
seed = (seed * 1103515245 + 12345) % 4294967296
return ((seed / 65536) % 32768)
}
/* Random number in range [from, to]. */
define rangerand(from, to) {
auto b, h, i, m, n, r
m = to - from + 1
h = length(m) / 2 + 1 /* want h iterations of rand() % 100 */
b = 100 ^ h % m /* want n >= b */
while (1) {
n = 0 /* pick n in range [b, 100 ^ h) */
for (i = h; i > 0; i--) {
r = rand()
while (r < 68) { r = rand(); } /* loop if the modulo bias */
n = (n * 100) + (r % 100) /* append 2 digits to n */
}
if (n >= b) { break; } /* break unless the modulo bias */
}
return (from + (n % m))
}
/* n is probably prime? */
define miller_rabin_test(n, k) {
auto d, r, a, x, s
if (n <= 3) { return (1); }
if ((n % 2) == 0) { return (0); }
/* find s and d so that d * 2^s = n - 1 */
d = n - 1
s = 0
while((d % 2) == 0) {
d /= 2
s += 1
}
while (k-- > 0) {
a = rangerand(2, n - 2)
x = (a ^ d) % n
if (x != 1) {
for (r = 0; r < s; r++) {
if (x == (n - 1)) { break; }
x = (x * x) % n
}
if (x != (n - 1)) {
return (0)
}
}
}
return (1)
}
for (i = 1; i < 1000; i++) {
if (miller_rabin_test(i, 10) == 1) {
i
}
}
quit

70
Task/Miller-Rabin-primality-test/Bracmat/miller-rabin-primality-test.bracmat
View file

@ -1,70 +0,0 @@
( 1:?seed
& ( rand
=
. mod$(!seed*1103515245+12345.4294967296):?seed
& mod$(div$(!seed.65536).32768)
)
& ( rangerand
= from to b h i m n r length
. !arg:(?from,?to)
& !to+-1*!from+1:?m
& @(!m:? [?length)
& div$(!length+1.2)+1:?h
& 100^mod$(!h.!m):?b
& whl
' ( 0:?n
& !h+1:?i
& whl
' ( !i+-1:>0:?i
& rand$:?r
& whl'(!r:<68&rand$:?r)
& !n*100+mod$(!r.100):?n
)
& !n:>!b
)
& !from+mod$(!n.!m)
)
& ( miller-rabin-test
= n k d r a x s return
. !arg:(?n,?k)
& ( !n:~>3&1
| mod$(!n.2):0
| !n+-1:?d
& 0:?s
& whl
' ( mod$(!d.2):0
& !d*1/2:?d
& 1+!s:?s
)
& 1:?return
& whl
' ( !k+-1:?k:~<0
& rangerand$(2,!n+-2):?a
& mod$(!a^!d.!n):?x
& ( !x:1
| 0:?r
& whl
' ( !r+1:~>!s:?r
& !n+-1:~!x
& mod$(!x*!x.!n):?x
)
& ( !n+-1:!x
| 0:?return&~
)
)
)
& !return
)
)
& 0:?i
& :?primes
& whl
' ( 1+!i:<1000:?i
& ( miller-rabin-test$(!i,10):1
& !primes !i:?primes
|
)
)
& !primes:? [-11 ?last
& out$!last
);

132
Task/Miller-Rabin-primality-test/C++/miller-rabin-primality-test.cpp
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@ -1,132 +0,0 @@
#include <algorithm>
#include <cmath>
#include <cstdint>
#include <iostream>
#include <vector>
std::vector<uint32_t> small_primes{ 2, 3 };
uint64_t add_modulus(const uint64_t& a, const uint64_t& b, const uint64_t& modulus) {
uint64_t am = ( a < modulus ) ? a : a % modulus;
if ( b == 0 ) {
return am;
}
uint64_t bm = ( b < modulus ) ? b : b % modulus;
uint64_t b_from_m = modulus - bm;
if ( am >= b_from_m ) {
return am - b_from_m;
}
return am + bm;
}
uint64_t multiply_modulus(const uint64_t& a, const uint64_t& b, const uint64_t& modulus) {
uint64_t am = ( a < modulus ) ? a : a % modulus;
uint64_t bm = ( b < modulus ) ? b : b % modulus;
if ( bm > am ) {
std::swap(am, bm);
}
uint64_t result;
while ( bm > 0 ) {
if ( ( bm & 1) == 1 ) {
result = add_modulus(result, am, modulus);
}
am = ( am << 1 ) - ( am >= ( modulus - am ) ? modulus : 0 );
bm >>= 1;
}
return result;
}
uint64_t exponentiation_modulus(const uint64_t& base, const uint64_t& exponent, const uint64_t& modulus) {
uint64_t b = base;
uint64_t e = exponent;
uint64_t result = 1;
while ( e > 0 ) {
if ( ( e & 1 ) == 1 ) {
result = multiply_modulus(result, b, modulus);
}
e >>= 1;
b = multiply_modulus(b, b, modulus);
}
return result;
}
bool is_composite(const uint32_t& a, const uint64_t& d, const uint64_t& n, const uint32_t& s) {
if ( exponentiation_modulus(a, d, n) == 1 ) {
return false;
}
for ( uint64_t i = 0; i < s; ++i ) {
if ( exponentiation_modulus(a, pow(2, i) * d, n) == n - 1 ) {
return false;
}
}
return true;
}
bool composite_test(const std::vector<uint32_t>& primes, const uint64_t& d, const uint64_t& n, const uint32_t& s) {
for ( const uint32_t& prime : primes ) {
if ( is_composite(prime, d, n, s) ) {
return true;
}
}
return false;
}
bool is_prime(const uint64_t& n) {
if ( n == 0 || n == 1 ) {
return false;
}
if ( std::find(small_primes.begin(), small_primes.end(), n) != small_primes.end() ) {
return true;
}
if ( std::any_of(small_primes.begin(), small_primes.end(), [n](uint32_t p) { return n % p == 0; }) ) {
return false;
}
uint64_t d = n - 1;
uint32_t s = 0;
while ( ! d % 2 ) {
d >>= 1;
s++;
}
if ( n < 1'373'653 ) {
return composite_test({ 2, 3 }, d, n, s);
}
if ( n < 25'326'001 ) {
return composite_test({ 2, 3, 5 }, d, n, s);
}
if ( n < 118'670'087'467 ) {
if ( n == 3'215'031'751 ) {
return false;
}
return composite_test({ 2, 3, 5, 7 }, d, n, s);
}
if ( n < 2'152'302'898'747 ) {
return composite_test({ 2, 3, 5, 7, 11 }, d, n, s);
}
if ( n < 3'474'749'660'383 ) {
return composite_test({ 2, 3, 5, 7, 11, 13 }, d, n, s);
}
if ( n < 341'550'071'728'321 ) {
return composite_test({ 2, 3, 5, 7, 11, 13, 17 }, d, n, s);
}
const std::vector<uint32_t> test_primes(small_primes.begin(), small_primes.begin() + 16);
return composite_test(test_primes, d, n, s);
}
void create_small_primes() {
for ( uint32_t i = 5; i < 1'000; i += 2 ) {
if ( is_prime(i) ) {
small_primes.emplace_back(i);
}
}
}
int main() {
create_small_primes();
for ( const uint64_t number : { 1'234'567'890'123'456'733, 1'234'567'890'123'456'737 } ) {
std::cout << "is_prime(" << number << ") = " << std::boolalpha << is_prime(number) << std::endl;
}
}

27
Task/Miller-Rabin-primality-test/C-sharp/miller-rabin-primality-test-1.cs
View file

@ -1,27 +0,0 @@
public static class RabinMiller
{
public static bool IsPrime(int n, int k)
{
if ((n < 2) || (n % 2 == 0)) return (n == 2);
int s = n - 1;
while (s % 2 == 0) s >>= 1;
Random r = new Random();
for (int i = 0; i < k; i++)
{
int a = r.Next(n - 1) + 1;
int temp = s;
long mod = 1;
for (int j = 0; j < temp; ++j) mod = (mod * a) % n;
while (temp != n - 1 && mod != 1 && mod != n - 1)
{
mod = (mod * mod) % n;
temp *= 2;
}
if (mod != n - 1 && temp % 2 == 0) return false;
}
return true;
}
}

58
Task/Miller-Rabin-primality-test/C-sharp/miller-rabin-primality-test-2.cs
View file

@ -1,58 +0,0 @@
// Miller-Rabin primality test as an extension method on the BigInteger type.
// Based on the Ruby implementation on this page.
public static class BigIntegerExtensions
{
public static bool IsProbablePrime(this BigInteger source, int certainty)
{
if(source == 2 || source == 3)
return true;
if(source < 2 || source % 2 == 0)
return false;
BigInteger d = source - 1;
int s = 0;
while(d % 2 == 0)
{
d /= 2;
s += 1;
}
// There is no built-in method for generating random BigInteger values.
// Instead, random BigIntegers are constructed from randomly generated
// byte arrays of the same length as the source.
RandomNumberGenerator rng = RandomNumberGenerator.Create();
byte[] bytes = new byte[source.ToByteArray().LongLength];
BigInteger a;
for(int i = 0; i < certainty; i++)
{
do
{
// This may raise an exception in Mono 2.10.8 and earlier.
// http://bugzilla.xamarin.com/show_bug.cgi?id=2761
rng.GetBytes(bytes);
a = new BigInteger(bytes);
}
while(a < 2 || a >= source - 2);
BigInteger x = BigInteger.ModPow(a, d, source);
if(x == 1 || x == source - 1)
continue;
for(int r = 1; r < s; r++)
{
x = BigInteger.ModPow(x, 2, source);
if(x == 1)
return false;
if(x == source - 1)
break;
}
if(x != source - 1)
return false;
}
return true;
}
}

5
Task/Miller-Rabin-primality-test/C/miller-rabin-primality-test-1.c
View file

@ -1,5 +0,0 @@
#ifndef _MILLER_RABIN_H_
#define _MILLER_RABIN_H
#include <gmp.h>
bool miller_rabin_test(mpz_t n, int j);
#endif

63
Task/Miller-Rabin-primality-test/C/miller-rabin-primality-test-2.c
View file

@ -1,63 +0,0 @@
#include <stdbool.h>
#include <gmp.h>
#include "primedecompose.h"
#define MAX_DECOMPOSE 100
bool miller_rabin_test(mpz_t n, int j)
{
bool res;
mpz_t f[MAX_DECOMPOSE];
mpz_t s, d, a, x, r;
mpz_t n_1, n_3;
gmp_randstate_t rs;
int l=0, k;
res = false;
gmp_randinit_default(rs);
mpz_init(s); mpz_init(d);
mpz_init(a); mpz_init(x); mpz_init(r);
mpz_init(n_1); mpz_init(n_3);
if ( mpz_cmp_si(n, 3) <= 0 ) { // let us consider 1, 2, 3 as prime
gmp_randclear(rs);
return true;
}
if ( mpz_odd_p(n) != 0 ) {
mpz_sub_ui(n_1, n, 1); // n-1
mpz_sub_ui(n_3, n, 3); // n-3
l = decompose(n_1, f);
mpz_set_ui(s, 0);
mpz_set_ui(d, 1);
for(k=0; k < l; k++) {
if ( mpz_cmp_ui(f[k], 2) == 0 )
mpz_add_ui(s, s, 1);
else
mpz_mul(d, d, f[k]);
} // 2^s * d = n-1
while(j-- > 0) {
mpz_urandomm(a, rs, n_3); // random from 0 to n-4
mpz_add_ui(a, a, 2); // random from 2 to n-2
mpz_powm(x, a, d, n);
if ( mpz_cmp_ui(x, 1) == 0 ) continue;
mpz_set_ui(r, 0);
while( mpz_cmp(r, s) < 0 ) {
if ( mpz_cmp(x, n_1) == 0 ) break;
mpz_powm_ui(x, x, 2, n);
mpz_add_ui(r, r, 1);
}
if ( mpz_cmp(x, n_1) == 0 ) continue;
goto flush; // woops
}
res = true;
}
flush:
for(k=0; k < l; k++) mpz_clear(f[k]);
mpz_clear(s); mpz_clear(d);
mpz_clear(a); mpz_clear(x); mpz_clear(r);
mpz_clear(n_1); mpz_clear(n_3);
gmp_randclear(rs);
return res;
}

29
Task/Miller-Rabin-primality-test/C/miller-rabin-primality-test-3.c
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@ -1,29 +0,0 @@
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <gmp.h>
#include "miller-rabin.h"
#define PREC 10
#define TOP 4000
int main()
{
mpz_t num;
mpz_init(num);
mpz_set_ui(num, 1);
while ( mpz_cmp_ui(num, TOP) < 0 ) {
if ( miller_rabin_test(num, PREC) ) {
gmp_printf("%Zd maybe prime\n", num);
} /*else {
gmp_printf("%Zd not prime\n", num);
}*/ // remove the comment iff you're interested in
// sure non-prime.
mpz_add_ui(num, num, 1);
}
mpz_clear(num);
return EXIT_SUCCESS;
}

72
Task/Miller-Rabin-primality-test/C/miller-rabin-primality-test-4.c
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@ -1,72 +0,0 @@
// calcul a^n%mod
size_t power(size_t a, size_t n, size_t mod)
{
size_t power = a;
size_t result = 1;
while (n)
{
if (n & 1)
result = (result * power) % mod;
power = (power * power) % mod;
n >>= 1;
}
return result;
}
// n−1 = 2^s * d with d odd by factoring powers of 2 from n−1
bool witness(size_t n, size_t s, size_t d, size_t a)
{
size_t x = power(a, d, n);
size_t y;
while (s) {
y = (x * x) % n;
if (y == 1 && x != 1 && x != n-1)
return false;
x = y;
--s;
}
if (y != 1)
return false;
return true;
}
/*
* if n < 1,373,653, it is enough to test a = 2 and 3;
* if n < 9,080,191, it is enough to test a = 31 and 73;
* if n < 4,759,123,141, it is enough to test a = 2, 7, and 61;
* if n < 1,122,004,669,633, it is enough to test a = 2, 13, 23, and 1662803;
* if n < 2,152,302,898,747, it is enough to test a = 2, 3, 5, 7, and 11;
* if n < 3,474,749,660,383, it is enough to test a = 2, 3, 5, 7, 11, and 13;
* if n < 341,550,071,728,321, it is enough to test a = 2, 3, 5, 7, 11, 13, and 17.
*/
bool is_prime_mr(size_t n)
{
if (((!(n & 1)) && n != 2 ) || (n < 2) || (n % 3 == 0 && n != 3))
return false;
if (n <= 3)
return true;
size_t d = n / 2;
size_t s = 1;
while (!(d & 1)) {
d /= 2;
++s;
}
if (n < 1373653)
return witness(n, s, d, 2) && witness(n, s, d, 3);
if (n < 9080191)
return witness(n, s, d, 31) && witness(n, s, d, 73);
if (n < 4759123141)
return witness(n, s, d, 2) && witness(n, s, d, 7) && witness(n, s, d, 61);
if (n < 1122004669633)
return witness(n, s, d, 2) && witness(n, s, d, 13) && witness(n, s, d, 23) && witness(n, s, d, 1662803);
if (n < 2152302898747)
return witness(n, s, d, 2) && witness(n, s, d, 3) && witness(n, s, d, 5) && witness(n, s, d, 7) && witness(n, s, d, 11);
if (n < 3474749660383)
return witness(n, s, d, 2) && witness(n, s, d, 3) && witness(n, s, d, 5) && witness(n, s, d, 7) && witness(n, s, d, 11) && witness(n, s, d, 13);
return witness(n, s, d, 2) && witness(n, s, d, 3) && witness(n, s, d, 5) && witness(n, s, d, 7) && witness(n, s, d, 11) && witness(n, s, d, 13) && witness(n, s, d, 17);
}

37
Task/Miller-Rabin-primality-test/C/miller-rabin-primality-test-5.c
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@ -1,37 +0,0 @@
typedef unsigned long long int ulong;
ulong mul_mod(ulong a, ulong b, const ulong mod) {
ulong res = 0, c; // return (a * b) % mod, avoiding overflow errors while doing modular multiplication.
for (b %= mod; a; a & 1 ? b >= mod - res ? res -= mod : 0, res += b : 0, a >>= 1, (c = b) >= mod - b ? c -= mod : 0, b += c);
return res % mod;
}
ulong pow_mod(ulong n, ulong exp, const ulong mod) {
ulong res = 1; // return (n ^ exp) % mod
for (n %= mod; exp; exp & 1 ? res = mul_mod(res, n, mod) : 0, n = mul_mod(n, n, mod), exp >>= 1);
return res;
}
int is_prime(ulong N) {
// Perform a Miller-Rabin test, it should be a deterministic version.
const ulong n_primes = 9, primes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23};
for (ulong i = 0; i < n_primes; ++i)
if (N % primes[i] == 0) return N == primes[i];
if (N < primes[n_primes - 1]) return 0;
int res = 1, s = 0;
ulong t;
for (t = N - 1; ~t & 1; t >>= 1, ++s);
for (ulong i = 0; i < n_primes && res; ++i) {
ulong B = pow_mod(primes[i], t, N);
if (B != 1) {
for (int b = s; b-- && (res = B + 1 != N);)
B = mul_mod(B, B, N);
res = !res;
}
}
return res;
}
int main(void){
return is_prime(8193145868754512737);
}

74
Task/Miller-Rabin-primality-test/Clojure/miller-rabin-primality-test-1.clj
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@ -1,74 +0,0 @@
(ns test-p.core
(:require [clojure.math.numeric-tower :as math])
(:require [clojure.set :as set]))
(def WITNESSLOOP "witness")
(def COMPOSITE "composite")
(defn m* [p q m]
" Computes (p*q) mod m "
(mod (*' p q) m))
(defn power
"modular exponentiation (i.e. b^e mod m"
[b e m]
(loop [b b, e e, x 1]
(if (zero? e)
x
(if (even? e) (recur (m* b b m) (quot e 2) x)
(recur (m* b b m) (quot e 2) (m* b x m))))))
; Sequence of random numbers to use in the test
(defn rand-num [n]
" random number between 2 and n-2 "
(bigint (math/floor (+' 2 (*' (- n 4) (rand))))))
; Unique set of random numbers
(defn unique-random-numbers [n k]
" k unique random numbers between 2 and n-2 "
(loop [a-set #{}]
(cond
(>= (count a-set) k) a-set
:else (recur (conj a-set (rand-num n))))))
(defn find-d-s [n]
" write n − 1 as 2s·d with d odd "
(loop [d (dec n), s 0]
(if (odd? d)
[d s]
(recur (quot d 2) (inc s)))))
(defn random-test
([n] (random-test n (min 1000 (bigint (/ n 2)))))
([n k]
" Random version of primality test"
(let [[d s] (find-d-s n)
; Individual Primality Test
single-test (fn [a s]
(let [z (power a d n)]
(if (some #{z} [1 (dec n)])
WITNESSLOOP
(loop [x (power z 2 n), r s]
(cond
(= x 1) COMPOSITE
(= x (dec n)) WITNESSLOOP
(= r 0) COMPOSITE
:else (recur (power x 2 n) (dec r)))))))]
; Apply Test
;(not-any? #(= COMPOSITE (local-test % s))
; (unique-random-numbers n k))))
(not-any? #(= COMPOSITE (single-test % s)) (unique-random-numbers n k)))))
;; Testing
(println "Primes beteen 900-1000:")
(doseq [q (range 900 1000)
:when (random-test q)]
(print " " q))
(println)
(println "Is Prime?" 4547337172376300111955330758342147474062293202868155909489 (random-test 4547337172376300111955330758342147474062293202868155909489))
(println "Is Prime?" 4547337172376300111955330758342147474062293202868155909393 (random-test 4547337172376300111955330758342147474062293202868155909393))
(println "Is Prime?" 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(random-test 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(random-test 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))

82
Task/Miller-Rabin-primality-test/Clojure/miller-rabin-primality-test-2.clj
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@ -1,82 +0,0 @@
(ns test-p.core
(:require [clojure.math.numeric-tower :as math]))
(def WITNESSLOOP "witness")
(def COMPOSITE "composite")
(defn m* [p q m]
" Computes (p*q) mod m "
(mod (*' p q) m))
(defn power
"modular exponentiation (i.e. b^e mod m"
[b e m]
(loop [b b, e e, x 1]
(if (zero? e)
x
(if (even? e) (recur (m* b b m) (quot e 2) x)
(recur (m* b b m) (quot e 2) (m* b x m))))))
(defn find-d-s [n]
" write n − 1 as 2s·d with d odd "
(loop [d (dec n), s 0]
(if (odd? d)
[d s]
(recur (quot d 2) (inc s)))))
;; Deterministic Test
(defn individual-deterministic-test [a d n s]
" Deterministic Primality Test "
(let [z (power a d n)]
(if (= z 1)
WITNESSLOOP
(loop [x z, r s]
(cond
(= x (dec n)) WITNESSLOOP
(zero? r) COMPOSITE
:else (recur (power x 2 n) (dec r)))))))
(defn deterministic-test [n]
" Sequence of Primality Tests "
(cond
(some #{n} [0 1 4]) false
(some #{n} [2 3]) true
(even? n) false
:else (let [[d s] (find-d-s n)]
(cond
(< n 2047) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 ])
(< n 1373653) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3])
(< n 9090191) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [31 73])
(< n 25326001) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5])
(< n 3215031751) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7])
(< n 1122004669633) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 13 23 1662803])
(< n 2152302898747) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11])
(< n 2152302898747) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11])
(< n 3474749660383) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13])
(< n 341550071728321) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17])
(< n 3825123056546,413,051) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23])
(< n (math/expt 2 64) ) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23 29 31 37])
(< n 318665857834031151167461) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23 29 31 37])
(< n 3317044064679887385961981) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23 29 31 37 41])
:else (let [k (min (dec n) (int (math/expt (Math/log n) 2)))]
(not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) (range 2 (inc k))))))))
;; Testing
(println "Primes beteen 900-1000:")
(doseq [q (range 900 1000)
:when (deterministic-test q)]
(print " " q))
(println)
(println "Is Prime?" 4547337172376300111955330758342147474062293202868155909489 (deterministic-test 4547337172376300111955330758342147474062293202868155909489))
(println "Is Prime?" 4547337172376300111955330758342147474062293202868155909393 (deterministic-test 4547337172376300111955330758342147474062293202868155909393))
println "Is Prime?" 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))

30
Task/Miller-Rabin-primality-test/Commodore-BASIC/miller-rabin-primality-test.basic
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@ -1,30 +0,0 @@
100 PRINT CHR$(147); CHR$(18); "**** MILLER-RABIN PRIMALITY TEST ****": PRINT
110 INPUT "NUMBER TO TEST"; N$
120 N = VAL(N$): IF N < 2 THEN 110
130 IF 0 = (N AND 1) THEN PRINT "(EVEN)": GOTO 380
140 INPUT "ITERATIONS"; K$
150 K = VAL(K$): IF K < 1 THEN 140
160 PRINT
170 DEF FNMD(M) = M - N * INT(M / N)
180 D = N - 1
190 S = 0
200 D = D / 2: S = S + 1
210 IF 0 = (D AND 1) THEN 200
220 P = 1
230 FOR I = 1 TO K
240 : A = INT(RND(.) * (N - 2)) + 2
250 : X = 1
260 : FOR J = 1 TO D
270 : X = FNMD(X * A)
280 : NEXT J
290 : IF (X = 1) OR (X = (N - 1)) THEN 360
300 : FOR J = 1 TO S - 1
310 : X = FNMD(X * X)
320 : IF X = 1 THEN P = 0: GOTO 370
330 : IF X = N - 1 THEN 360
340 : NEXT J
350 : P = 0: GOTO 370
360 NEXT I
370 P = P * (1 - 1 / (4 * K))
380 IF P THEN PRINT "PROBABLY PRIME ( P >="; P; ")": END
390 PRINT "COMPOSITE."

43
Task/Miller-Rabin-primality-test/Common-Lisp/miller-rabin-primality-test.lisp
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@ -1,43 +0,0 @@
(defun factor-out (number divisor)
"Return two values R and E such that NUMBER = DIVISOR^E * R,
and R is not divisible by DIVISOR."
(do ((e 0 (1+ e))
(r number (/ r divisor)))
((/= (mod r divisor) 0) (values r e))))
(defun mult-mod (x y modulus) (mod (* x y) modulus))
(defun expt-mod (base exponent modulus)
"Fast modular exponentiation by repeated squaring."
(labels ((expt-mod-iter (b e p)
(cond ((= e 0) p)
((evenp e)
(expt-mod-iter (mult-mod b b modulus)
(/ e 2)
p))
(t
(expt-mod-iter b
(1- e)
(mult-mod b p modulus))))))
(expt-mod-iter base exponent 1)))
(defun random-in-range (lower upper)
"Return a random integer from the range [lower..upper]."
(+ lower (random (+ (- upper lower) 1))))
(defun miller-rabin-test (n k)
"Test N for primality by performing the Miller-Rabin test K times.
Return NIL if N is composite, and T if N is probably prime."
(cond ((= n 1) nil)
((< n 4) t)
((evenp n) nil)
(t
(multiple-value-bind (d s) (factor-out (- n 1) 2)
(labels ((strong-liar? (a)
(let ((x (expt-mod a d n)))
(or (= x 1)
(loop repeat s
for y = x then (mult-mod y y n)
thereis (= y (- n 1)))))))
(loop repeat k
always (strong-liar? (random-in-range 2 (- n 2)))))))))

40
Task/Miller-Rabin-primality-test/Crystal/miller-rabin-primality-test-1.cr
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@ -1,40 +0,0 @@
require "big"
module Primes
module MillerRabin
def prime?(k = 15) # increase k for more confidence
neg_one_mod = d = self - 1
s = 0
while d.even?; d >>= 1; s += 1 end # d is odd after s shifts
k.times do
b = 2 + rand(self - 4) # random witness base b
y = powmod(b, d, self) # y = (b**d) mod self
next if y == 1 || y == neg_one_mod
(s - 1).times do
y = (y * y) % self # y = (y**2) mod self
return false if y == 1
break if y == neg_one_mod
end
return false if y != neg_one_mod
end
true # prime (with high probability)
end
# Compute b**e mod m
private def powmod(b, e, m)
r, b = 1, b.to_big_i
while e > 0
r = (b * r) % m if e.odd?
b = (b * b) % m
e >>= 1
end
r
end
end
end
struct Int; include Primes::MillerRabin end
puts 341521.prime?(20) # => true
puts 341531.prime? # => false

144
Task/Miller-Rabin-primality-test/Crystal/miller-rabin-primality-test-2.cr
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@ -1,144 +0,0 @@
# For crystal >= 0.31.x, compile without overflow check, as either
# crystal build miller-rabin.cr -Ddisable_overflow --release
# crystal build -Ddisable_overflow miller-rabin.cr --release
require "big"
module Primes
module MillerRabin
# Returns true if +self+ is a prime number, else returns false.
def primemr?(k = 10)
primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}
return primes.includes? self if self <= primes.last
modp47 = 614_889_782_588_491_410.to_big_i # => primes.product, largest < 2^64
return false if modp47.gcd(self.to_big_i) != 1 # eliminates 86.2% of all integers
# Choose input witness bases: wits = [range, [wit_bases]] or nil
wits = WITNESS_RANGES.find { |range, wits| range > self }
witnesses = wits && wits[1] || k.times.map{ 2 + rand(self - 4) }
miller_rabin_test(witnesses)
end
# Returns true if +self+ passes Miller-Rabin Test on witnesses +b+
private def miller_rabin_test(witnesses) # list of witnesses for testing
neg_one_mod = n = d = self - 1 # these are even as self is always odd
d >>= d.trailing_zeros_count # shift out factors of 2 to make d odd
witnesses.each do |b| # do M-R test with each witness base
next if (b % self) == 0 # **skip base if a multiple of input**
y = powmod(b, d, self) # y = (b**d) mod self
s = d
until y == 1 || y == neg_one_mod || s == n
y = (y * y) % self # y = (y**2) mod self
s <<= 1
end
return false unless y == neg_one_mod || s.odd?
end
true
end
# Best known deterministic witnesses for given range and set of bases
# https://miller-rabin.appspot.com/
# https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
private WITNESS_RANGES = {
341_531 => {9345883071009581737},
1_050_535_501 => {336781006125, 9639812373923155},
350_269_456_337 => {4230279247111683200, 14694767155120705706, 16641139526367750375},
55_245_642_489_451 => {2, 141889084524735, 1199124725622454117, 11096072698276303650},
7_999_252_175_582_851 => {2, 4130806001517, 149795463772692060, 186635894390467037,
3967304179347715805},
585_226_005_592_931_977 => {2, 123635709730000, 9233062284813009, 43835965440333360,
761179012939631437, 1263739024124850375},
18_446_744_073_709_551_615 => {2, 325, 9375, 28178, 450775, 9780504, 1795265022},
"318665857834031151167461".to_big_i => {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37},
"3317044064679887385961981".to_big_i => {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41}
}
# Compute b**e mod m
private def powmod(b, e, m)
r, b = 1, b.to_big_i
while e > 0
r = (b * r) % m if e.odd?
b = (b * b) % m
e >>= 1
end
r
end
end
end
struct Int; include Primes::MillerRabin end
def tm; t = Time.monotonic; yield; (Time.monotonic - t).total_seconds.round(6) end
# 10 digit prime
n = 2147483647
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 18 digit non-prime
n = 844674407370955389
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 19 digit prime
n = 9241386435364257883.to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 20 digit prime; largest < 2^64
n = 18446744073709551533.to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 58 digit prime
n = "4547337172376300111955330758342147474062293202868155909489".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 58 digit non-prime
n = "4547337172376300111955330758342147474062293202868155909393".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 81 digit prime
n = "100000000000000000000000000000000000000000000000000000000000000000000000001309503".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 81 digit non-prime
n = "100000000000000000000000000000000000000000000000000000000000000000000000001309509".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 308 digit prime
n = "94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = "138028649176899647846076023812164793645371887571371559091892986639999096471811910222267538577825033963552683101137782650479906670021895135954212738694784814783986671046107023185842481502719762055887490765764329237651328922972514308635045190654896041748716218441926626988737664133219271115413563418353821396401".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = "123301261697053560451930527879636974557474268923771832437126939266601921428796348203611050423256894847735769138870460373141723679005090549101566289920247264982095246187318303659027201708559916949810035265951104246512008259674244307851578647894027803356820480862664695522389066327012330793517771435385653616841".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = "119432521682023078841121052226157857003721669633106050345198988740042219728400958282159638484144822421840470442893056822510584029066504295892189315912923804894933736660559950053226576719285711831138657839435060908151231090715952576998400120335346005544083959311246562842277496260598128781581003807229557518839".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = "132082885240291678440073580124226578272473600569147812319294626601995619845059779715619475871419551319029519794232989255381829366374647864619189704922722431776563860747714706040922215308646535910589305924065089149684429555813953571007126408164577035854428632242206880193165045777949624510896312005014225526731".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = "153410708946188157980279532372610756837706984448408515364579602515073276538040155990230789600191915021209039203172105094957316552912585741177975853552299222501069267567888742458519569317286299134843250075228359900070009684517875782331709619287588451883575354340318132216817231993558066067063143257425853927599".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = "103130593592068072608023213244858971741946977638988649427937324034014356815504971087381663169829571046157738503075005527471064224791270584831779395959349442093395294980019731027051356344056416276026592333932610954020105156667883269888206386119513058400355612571198438511950152690467372712488391425876725831041".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = "94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881".to_big_i
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts

50
Task/Miller-Rabin-primality-test/D/miller-rabin-primality-test.d
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@ -1,50 +0,0 @@
import std.random;
bool isProbablePrime(in ulong n, in uint k=10) /*nothrow*/ @safe /*@nogc*/ {
static ulong modPow(ulong b, ulong e, in ulong m)
pure nothrow @safe @nogc {
ulong result = 1;
while (e > 0) {
if ((e & 1) == 1)
result = (result * b) % m;
b = (b ^^ 2) % m;
e >>= 1;
}
return result;
}
if (n < 2 || n % 2 == 0)
return n == 2;
ulong d = n - 1;
ulong s = 0;
while (d % 2 == 0) {
d /= 2;
s++;
}
assert(2 ^^ s * d == n - 1);
outer:
foreach (immutable _; 0 .. k) {
immutable ulong a = uniform(2, n);
ulong x = modPow(a, d, n);
if (x == 1 || x == n - 1)
continue;
foreach (immutable __; 1 .. s) {
x = modPow(x, 2, n);
if (x == 1)
return false;
if (x == n - 1)
continue outer;
}
return false;
}
return true;
}
void main() { // Demo code.
import std.stdio, std.range, std.algorithm;
iota(2, 30).filter!isProbablePrime.writeln;
}

23
Task/Miller-Rabin-primality-test/E/miller-rabin-primality-test-1.e
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@ -1,23 +0,0 @@
def millerRabinPrimalityTest(n :(int > 0), k :int, random) :boolean {
if (n <=> 2 || n <=> 3) { return true }
if (n <=> 1 || n %% 2 <=> 0) { return false }
var d := n - 1
var s := 0
while (d %% 2 <=> 0) {
d //= 2
s += 1
}
for _ in 1..k {
def nextTrial := __continue
def a := random.nextInt(n - 3) + 2 # [2, n - 2] = [0, n - 4] + 2 = [0, n - 3) + 2
var x := a**d %% n # Note: Will do optimized modular exponentiation
if (x <=> 1 || x <=> n - 1) { nextTrial() }
for _ in 1 .. (s - 1) {
x := x**2 %% n
if (x <=> 1) { return false }
if (x <=> n - 1) { nextTrial() }
}
return false
}
return true
}

4
Task/Miller-Rabin-primality-test/E/miller-rabin-primality-test-2.e
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@ -1,4 +0,0 @@
for i ? (millerRabinPrimalityTest(i, 1, entropy)) in 4..1000 {
print(i, " ")
}
println()

39
Task/Miller-Rabin-primality-test/EchoLisp/miller-rabin-primality-test.l
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@ -1,39 +0,0 @@
(lib 'bigint)
;; output : #t if n probably prime
(define (miller-rabin n (k 7) (composite #f)(x))
(define d (1- n))
(define s 0)
(define a 0)
(while (even? d)
(set! s (1+ s))
(set! d (quotient d 2)))
(for [(i k)]
(set! a (+ 2 (random (- n 3))))
(set! x (powmod a d n))
#:continue (or (= x 1) (= x (1- n)))
(set! composite
(for [(r (in-range 1 s))]
(set! x (powmod x 2 n))
#:break (= x 1) => #t
#:break (= x (1- n)) => #f
#t
))
#:break composite => #f )
(not composite))
;; output
(miller-rabin #101)
→ #t
(miller-rabin #111)
→ #f
(define big-prime (random-prime 1e+100))
3461396142610375479080862553800503306376298093021233334170610435506057862777898396429
6627816219192601527
(miller-rabin big-prime)
→ #t
(miller-rabin (1+ (factorial 100)))
→ #f
(prime? (1+ (factorial 100))) ;; native
→ #f

49
Task/Miller-Rabin-primality-test/Elixir/miller-rabin-primality-test-1.ex
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@ -1,49 +0,0 @@
defmodule Prime do
use Application
alias :math, as: Math
alias :rand, as: Rand
def start( _type, _args ) do
primes = 5..1000
|> Enum.filter( fn( x ) -> (rem x, 2) == 1 end )
|> Enum.filter( fn( x ) -> miller_rabin?( x, 10) == True end )
IO.inspect( primes, label: "Primes: ", limit: :infinity )
{ :ok, self() }
end
def modular_exp( x, y, mod ) do
with [ _ | bits ] = Integer.digits( y, 2 ) do
Enum.reduce bits, x, fn( bit, acc ) -> acc * acc |> ( &( if bit == 1, do: &1 * x, else: &1 ) ).() |> rem( mod ) end
end
end
def miller_rabin( d, s ) when rem( d, 2 ) == 0, do: { s, d }
def miller_rabin( d, s ), do: miller_rabin( div( d, 2 ), s + 1 )
def miller_rabin?( n, g ) do
{ s, d } = miller_rabin( n - 1, 0 )
miller_rabin( n, g, s, d )
end
def miller_rabin( n, 0, _, _ ), do: True
def miller_rabin( n, g, s, d ) do
a = 1 + Rand.uniform( n - 3 )
x = modular_exp( a, d, n )
if x == 1 or x == n - 1 do
miller_rabin( n, g - 1, s, d )
else
if miller_rabin( n, x, s - 1) == True, do: miller_rabin( n, g - 1, s, d ), else: False
end
end
def miller_rabin( n, x, r ) when r <= 0, do: False
def miller_rabin( n, x, r ) do
x = modular_exp( x, 2, n )
unless x == 1 do
unless x == n - 1, do: miller_rabin( n, x, r - 1 ), else: True
else
False
end
end
end

7
Task/Miller-Rabin-primality-test/Elixir/miller-rabin-primality-test-2.ex
View file

@ -1,7 +0,0 @@
miller_rabin?( 94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881, 1000 )
miller_rabin?( 138028649176899647846076023812164793645371887571371559091892986639999096471811910222267538577825033963552683101137782650479906670021895135954212738694784814783986671046107023185842481502719762055887490765764329237651328922972514308635045190654896041748716218441926626988737664133219271115413563418353821396401, 1000 )
miller_rabin?( 123301261697053560451930527879636974557474268923771832437126939266601921428796348203611050423256894847735769138870460373141723679005090549101566289920247264982095246187318303659027201708559916949810035265951104246512008259674244307851578647894027803356820480862664695522389066327012330793517771435385653616841, 1000 )
miller_rabin?( 119432521682023078841121052226157857003721669633106050345198988740042219728400958282159638484144822421840470442893056822510584029066504295892189315912923804894933736660559950053226576719285711831138657839435060908151231090715952576998400120335346005544083959311246562842277496260598128781581003807229557518839, 1000 )
miller_rabin?( 132082885240291678440073580124226578272473600569147812319294626601995619845059779715619475871419551319029519794232989255381829366374647864619189704922722431776563860747714706040922215308646535910589305924065089149684429555813953571007126408164577035854428632242206880193165045777949624510896312005014225526731, 1000 )
miller_rabin?( 153410708946188157980279532372610756837706984448408515364579602515073276538040155990230789600191915021209039203172105094957316552912585741177975853552299222501069267567888742458519569317286299134843250075228359900070009684517875782331709619287588451883575354340318132216817231993558066067063143257425853927599, 1000 )
miller_rabin?( 103130593592068072608023213244858971741946977638988649427937324034014356815504971087381663169829571046157738503075005527471064224791270584831779395959349442093395294980019731027051356344056416276026592333932610954020105156667883269888206386119513058400355612571198438511950152690467372712488391425876725831041, 1000 )

90
Task/Miller-Rabin-primality-test/Erlang/miller-rabin-primality-test-1.erl
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@ -1,90 +0,0 @@
-module(miller_rabin).
-export([is_prime/1, power/2]).
is_prime(1) -> false;
is_prime(2) -> true;
is_prime(3) -> true;
is_prime(N) when N > 3, ((N rem 2) == 0) -> false;
is_prime(N) when ((N rem 2) ==1), N < 341550071728321 ->
is_mr_prime(N, proving_bases(N));
is_prime(N) when ((N rem 2) == 1) ->
is_mr_prime(N, random_bases(N, 100)).
proving_bases(N) when N < 1373653 ->
[2, 3];
proving_bases(N) when N < 9080191 ->
[31, 73];
proving_bases(N) when N < 25326001 ->
[2, 3, 5];
proving_bases(N) when N < 3215031751 ->
[2, 3, 5, 7];
proving_bases(N) when N < 4759123141 ->
[2, 7, 61];
proving_bases(N) when N < 1122004669633 ->
[2, 13, 23, 1662803];
proving_bases(N) when N < 2152302898747 ->
[2, 3, 5, 7, 11];
proving_bases(N) when N < 3474749660383 ->
[2, 3, 5, 7, 11, 13];
proving_bases(N) when N < 341550071728321 ->
[2, 3, 5, 7, 11, 13, 17].
is_mr_prime(N, As) when N>2, N rem 2 == 1 ->
{D, S} = find_ds(N),
%% this is a test for compositeness; the two case patterns disprove
%% compositeness.
not lists:any(fun(A) ->
case mr_series(N, A, D, S) of
[1|_] -> false; % first elem of list = 1
L -> not lists:member(N-1, L) % some elem of list = N-1
end
end,
As).
find_ds(N) ->
find_ds(N-1, 0).
find_ds(D, S) ->
case D rem 2 == 0 of
true ->
find_ds(D div 2, S+1);
false ->
{D, S}
end.
mr_series(N, A, D, S) when N rem 2 == 1 ->
Js = lists:seq(0, S),
lists:map(fun(J) -> pow_mod(A, power(2, J)*D, N) end, Js).
pow_mod(B, E, M) ->
case E of
0 -> 1;
_ -> case ((E rem 2) == 0) of
true -> (power(pow_mod(B, (E div 2), M), 2)) rem M;
false -> (B*pow_mod(B, E-1, M)) rem M
end
end.
random_bases(N, K) ->
[basis(N) || _ <- lists:seq(1, K)].
basis(N) when N>2 ->
1 + random:uniform(N-3). % random:uniform returns a single random number in range 1 -> N-3, to which is added 1, shifting the range to 2 -> N-2
power(B, E) ->
power(B, E, 1).
power(_, 0, Acc) ->
Acc;
power(B, E, Acc) ->
power(B, E - 1, B * Acc).

182
Task/Miller-Rabin-primality-test/Erlang/miller-rabin-primality-test-2.erl
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@ -1,182 +0,0 @@
%%% @author Tony Wallace <tony@resurrection>
%%% @copyright (C) 2021, Tony Wallace
%%% @doc
%%% For details of the algorithms used see
%%% https://en.wikipedia.org/wiki/Modular_exponentiation
%%% @end
%%% Created : 21 Jul 2021 by Tony Wallace <tony@resurrection>
-module mod.
-export [mod_mult/3,mod_exp/3,binary_exp/2,test/0].
mod_mult(I1,I2,Mod) when
I1 > Mod,
is_integer(I1), is_integer(I2), is_integer(Mod) ->
mod_mult(I1 rem Mod,I2,Mod);
mod_mult(I1,I2,Mod) when
I2 > Mod,
is_integer(I1), is_integer(I2), is_integer(Mod) ->
mod_mult(I1,I2 rem Mod,Mod);
mod_mult(I1,I2,Mod) when
is_integer(I1), is_integer(I2), is_integer(Mod) ->
(I1 * I2) rem Mod.
mod_exp(Base,Exp,Mod) when
is_integer(Base),
is_integer(Exp),
is_integer(Mod),
Base > 0,
Exp > 0,
Mod > 0 ->
binary_exp_mod(Base,Exp,Mod);
mod_exp(_,0,_) -> 1.
binary_exp(Base,Exponent) when
is_integer(Base),
is_integer(Exponent),
Base > 0,
Exponent > 0 ->
binary_exp(Base,Exponent,1);
binary_exp(_,0) ->
1.
binary_exp(_,0,Result) ->
Result;
binary_exp(Base,Exponent,Acc) ->
binary_exp(Base*Base,Exponent bsr 1,Acc * exp_factor(Base,Exponent)).
binary_exp_mod(Base,Exponent,Mod) ->
binary_exp_mod(Base rem Mod,Exponent,Mod,1).
binary_exp_mod(_,0,_,Result) ->
Result;
binary_exp_mod(Base,Exponent,Mod,Acc) ->
binary_exp_mod((Base*Base) rem Mod,
Exponent bsr 1,Mod,(Acc * exp_factor(Base,Exponent))rem Mod).
exp_factor(_,0) ->
1;
exp_factor(Base,1) ->
Base;
exp_factor(Base,Exponent) ->
exp_factor(Base,Exponent band 1).
test() ->
445 = mod_exp(4,13,497),
%% Rosetta code example:
R = 1527229998585248450016808958343740453059 =
mod_exp(2988348162058574136915891421498819466320163312926952423791023078876139,
2351399303373464486466122544523690094744975233415544072992656881240319,
binary_exp(10,40)),
R.
% mod module ends here
%% Modified version of rosetta code entry
%% Modification was more efficient exponentiation
%% Modification - use of rpc:pmap to utilise multithreaded CPUs
-module(miller_rabin).
-export([is_prime/1,mr_series_test/4,mersennes/1,test/0]).
is_prime(1) -> false;
is_prime(2) -> true;
is_prime(3) -> true;
is_prime(N) when N > 3, ((N rem 2) == 0) -> false;
is_prime(N) when ((N rem 2) ==1), N < 341550071728321 ->
is_mr_prime(N, proving_bases(N));
is_prime(N) when ((N rem 2) == 1) ->
is_mr_prime(N, random_bases(N, 100)).
proving_bases(N) when N < 1373653 ->
[2, 3];
proving_bases(N) when N < 9080191 ->
[31, 73];
proving_bases(N) when N < 25326001 ->
[2, 3, 5];
proving_bases(N) when N < 3215031751 ->
[2, 3, 5, 7];
proving_bases(N) when N < 4759123141 ->
[2, 7, 61];
proving_bases(N) when N < 1122004669633 ->
[2, 13, 23, 1662803];
proving_bases(N) when N < 2152302898747 ->
[2, 3, 5, 7, 11];
proving_bases(N) when N < 3474749660383 ->
[2, 3, 5, 7, 11, 13];
proving_bases(N) when N < 341550071728321 ->
[2, 3, 5, 7, 11, 13, 17].
is_mr_prime(N, As) when N>2, N rem 2 == 1 ->
% TStart = erlang:monotonic_time(),
{D, S} = find_ds(N),
% elapsed(TStart,"find_ds took ~p.~p seconds~n"),
%% this is a test for compositeness; the two case patterns disprove
%% compositeness.
TestResults =
rpc:pmap({miller_rabin,mr_series_test},[N,D,S],As),
R= not lists:any(fun(X) -> X end,TestResults),
% elapsed(TStart,"is_mr_prime took ~p.~p seconds~n"),
R.
mr_series_test(A,N,D,S) ->
% TMrS = erlang:monotonic_time(),
R = case mr_series(N, A, D, S) of
[1|_] -> false; % first elem of list = 1
L -> not lists:member(N-1, L) % some elem of list = N-1
end,
% elapsed(TMrS,"mr_series took ~p.~p seconds~n"),
R.
%elapsed(TStart,Msg) ->
% TElapsed_ms = erlang:convert_time_unit(erlang:monotonic_time()-TStart,native,1000),
% TSec = TElapsed_ms div 1000,
% Tms = TElapsed_ms rem 1000,
% io:format(Msg, [TSec,Tms]).
find_ds(N) ->
find_ds(N-1, 0).
find_ds(D, S) ->
case D rem 2 == 0 of
true ->
find_ds(D div 2, S+1);
false ->
{D, S}
end.
mr_series(N, A, D, S) when N rem 2 == 1 ->
Js = lists:seq(0, S),
lists:map(fun(J) -> mod:mod_exp(A, mod:binary_exp(2, J)*D, N) end, Js).
random_bases(N, K) ->
[basis(N) || _ <- lists:seq(1, K)].
basis(N) when N>2 ->
% random:uniform returns a single random number in range 1 -> N-3,
% to which is added 1, shifting the range to 2 -> N-2
1 + rand:uniform(N-3).
mersennes(N) when N>0, is_integer(N) ->
1 bsl N - 1.
test() ->
TStart = erlang:monotonic_time(),
true = is_prime(7),
true = is_prime(41),
false = is_prime(42),
true = is_prime(mersennes(31)),
true = is_prime(mersennes(127)), % M(127) checks okay if 64 bit word size exceeded,
true = is_prime(mersennes(3217)), % about the size of an rsa key,
TFinish = erlang:monotonic_time(),
ElapsedSeconds = erlang:convert_time_unit(TFinish - TStart,native,1),
io:format("Time seconds = ~p~n",[ElapsedSeconds]),
ok
.

11
Task/Miller-Rabin-primality-test/F-Sharp/miller-rabin-primality-test.fs
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@ -1,11 +0,0 @@
// Miller primality test for n<3317044064679887385961981. Nigel Galloway: April 1st., 2021
let a=[(2047I,[2I]);(1373653I,[2I;3I]);(9080191I,[31I;73I]);(25326001I,[2I;3I;5I]);(3215031751I,[2I;3I;5I;7I]);(4759123141I,[2I;7I;61I]);(1122004669633I,[2I;13I;23I;1662803I]);
(2152302898747I,[2I;3I;5I;7I;11I]);(3474749660383I,[2I;3I;5I;7I;11I;13I]);(341550071728321I,[2I;3I;5I;7I;11I;13I;17I]);(3825123056546413051I,[2I;3I;5I;7I;11I;13I;17I;19I;23I]);
(18446744073709551616I,[2I;3I;5I;7I;11I;13I;17I;19I;23I;29I;31I;37I]);(318665857834031151167461I,[2I;3I;5I;7I;11I;13I;17I;19I;23I;29I;31I;37I]);(3317044064679887385961981I,[2I;3I;5I;7I;11I;13I;17I;19I;23I;29I;31I;37I;41I])]
let rec fN g=function (n:bigint) when n.IsEven->fN(g+1)(n/2I) |n->(n,g)
let rec fG n d r=function a::t->match bigint.ModPow(a,d,n) with g when g=1I || g=n-1I->fG n d r t |g->fL(bigint.ModPow(g,2I,n)) n d t r
|_->true
and fL x n d a=function 1->false |r when x=n-1I->fG n d r a |r->fL(bigint.ModPow(x,2I,n)) n d a (r-1)
let mrP n=let (d,r)=fN 0 (n-1I) in fG n d r (snd(a|>List.find(fst>>(<)n)))
printfn "%A %A" (mrP 2147483647I)(mrP 844674407370955389I)

88
Task/Miller-Rabin-primality-test/Forth/miller-rabin-primality-test.fth
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@ -1,88 +0,0 @@
\ modular multiplication and exponentiation
\
: 3rd s" 2 pick" evaluate ; immediate
: mod* ( a b m -- a*b {mod m} )
>r um* r> ud/mod 2drop ;
: mod^ ( x n m -- x^n {mod m} )
>r 1 swap
begin ?dup while
dup 1 and 1 =
if
swap 3rd r@ mod* swap 1-
then dup 0>
if
rot dup r@ mod* -rot 2/
then
repeat nip rdrop ;
\ small divisor check: true => possibly prime; false => definitely not prime.
\
31 constant π-128
create maybe-prime?
2 c, 3 c, 5 c, 7 c, 11 c, 13 c, 17 c, 19 c, 23 c, 29 c,
31 c, 37 c, 41 c, 43 c, 47 c, 53 c, 59 c, 61 c, 67 c, 71 c,
73 c, 79 c, 83 c, 89 c, 97 c, 101 c, 103 c, 107 c, 109 c, 113 c,
127 c,
does>
true -rot
π-128 bounds do
i c@ dup * over > if leave then
dup i c@ mod 0= if 2drop false unloop exit then
loop drop ;
\ actual Miller-Rabin test
\
: factor-2s ( n -- s d )
0 swap
begin dup 1 and 0= while
swap 1+ swap 2/
repeat ;
: fermat-square-test ( n m s -- ? ) \ perform n = n^2 (mod m), s-1 times
1- 0 ?do
2dup - -1 =
if leave
then >r dup r@ mod* r>
loop
- -1 = ;
: strong-fermat-pseudoprime? ( n a -- ? )
over >r \ keep the modulus on the return stack
>r 1- factor-2s r> \ -- s d a
swap r@ mod^ \ s d a -- s, a^d (mod n)
dup 1 = \ a^d == 1 (mod n) => Fermat pseudoprime
if 2drop rdrop true
else r> rot fermat-square-test
then ;
4.759.123.141 drop constant mr-det-3 \ Deterministic threshold; 3 bases
create small-prime-bases 2 , 7 , 61 , \ deterministic up to mr-det-3
create large-prime-bases 2 , 325 , 9375 , 28178 , 450775 , 9780504 , 1795265022 , \ known to be deterministic for 64 bit integers.
: miler-rabin-bases ( n -- addr n )
mr-det-3 <
if small-prime-bases 3
else large-prime-bases 7
then ;
: miller-rabin-primality-test ( n -- f )
dup miler-rabin-bases cells bounds do
dup i @ strong-fermat-pseudoprime? invert
if drop false unloop exit then
cell +loop drop true ;
: prime? ( n -- f )
dup 2 <
if drop false
else
dup maybe-prime?
if dup [ 127 dup * 1+ ] literal <
if drop true
else miller-rabin-primality-test
then
else drop false
then
then ;

56
Task/Miller-Rabin-primality-test/Fortran/miller-rabin-primality-test-1.f
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@ -1,56 +0,0 @@
module Miller_Rabin
use PrimeDecompose
implicit none
integer, parameter :: max_decompose = 100
private :: int_rrand, max_decompose
contains
function int_rrand(from, to)
integer(huge) :: int_rrand
integer(huge), intent(in) :: from, to
real :: o
call random_number(o)
int_rrand = floor(from + o * real(max(from,to) - min(from, to)))
end function int_rrand
function miller_rabin_test(n, k) result(res)
logical :: res
integer(huge), intent(in) :: n
integer, intent(in) :: k
integer(huge), dimension(max_decompose) :: f
integer(huge) :: s, d, i, a, x, r
res = .true.
f = 0
if ( (n <= 2) .and. (n > 0) ) return
if ( mod(n, 2) == 0 ) then
res = .false.
return
end if
call find_factors(n-1, f)
s = count(f == 2)
d = (n-1) / (2 ** s)
loop: do i = 1, k
a = int_rrand(2_huge, n-2)
x = mod(a ** d, n)
if ( x == 1 ) cycle
do r = 0, s-1
if ( x == ( n - 1 ) ) cycle loop
x = mod(x*x, n)
end do
if ( x == (n-1) ) cycle
res = .false.
return
end do loop
res = .true.
end function miller_rabin_test
end module Miller_Rabin

24
Task/Miller-Rabin-primality-test/Fortran/miller-rabin-primality-test-2.f
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program TestMiller
use Miller_Rabin
implicit none
integer, parameter :: prec = 30
integer(huge) :: i
! this is limited since we're not using a bignum lib
call do_test( (/ (i, i=1, 29) /) )
contains
subroutine do_test(a)
integer(huge), dimension(:), intent(in) :: a
integer :: i
do i = 1, size(a,1)
print *, a(i), miller_rabin_test(a(i), prec)
end do
end subroutine do_test
end program TestMiller

77
Task/Miller-Rabin-primality-test/Fortran/miller-rabin-primality-test-3.f
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MODULE MRTEST !Try the Miller-Rabin primality test.
CONTAINS !Working only with in-built integers.
LOGICAL FUNCTION MRPRIME(N,TRIALS) !Could N be a prime number?
USE DFPORT !To get RAND.
INTEGER N !The number.
INTEGER TRIALS !The count of trials to make.
INTEGER D,S !Represents a number in a special form.
INTEGER TRIAL
INTEGER A,X,R
Catch some annoying cases.
IF (N .LE. 4) THEN !A single-digit number?
MRPRIME = N.GT.1 .AND. N.LE.3 !Yes. Some special values.
RETURN !Thus allow 2 to be reported as prime.
END IF !Yet, test for 2 as a possible factor for larger numbers.
MRPRIME = .FALSE. !Pessimism prevails.
IF (MOD(N,2).EQ.0 .OR. MOD(N,3).EQ.0) RETURN !Thus.
Construct D such that N - 1 = D*2**S. By here, N is odd, and greater than three.
D = N - 1 !Thus, D becomes an even number.
S = 1 !So, it has at least one power of two.
10 D = D/2 !Divide it out.
IF (MOD(D,2).EQ.0) THEN !If there is another,
S = S + 1 !Count it,
GO TO 10 !And divide it out also.
END IF !So, D is no longer even. N = 1 + D*2**S
WRITE (6,11) N,D,S
11 FORMAT("For ",I0,", D=",I0,",S=",I0)
Convince through repetition..
T:DO TRIAL = 1,TRIALS !Some trials yield a definite result.
A = RAND(0)*(N - 2) + 2 !For small N, the birthday problem.
X = MODEXP(N,A,D) !A**D mod N.
WRITE (6,22) TRIAL,A,X,INT8(A)**D,N,MOD(INT8(A)**D,N)
22 FORMAT(6X,"Trial ",I0,",A=",I4,",X=",I4,
1 "=MOD(",I0,",",I0,")=",I0)
IF (X.EQ.1 .OR. X.EQ.N - 1) CYCLE T !Pox. A prime yields these.
DO R = 1,S - 1 !Step through the powers of two in N - 1.
X = MODEXP(N,X,2) !X**2 mod N.
WRITE (6,23) R,X
23 FORMAT (14X,"R=",I4,",X=",I0)
IF (X.EQ.1) RETURN !Definitely composite. No prime does this.
IF (X.EQ.N - 1) CYCLE T !Pox. Try something else.
END DO !Another power of two?
RETURN !Definitely composite.
END DO T !Have another go.
MRPRIME = .TRUE. !Would further trials yield greater assurance?
END FUNCTION MRPRIME !Are some numbers resistant to this scheme?
INTEGER FUNCTION MODEXP(N,X,P) !Calculate X**P mod N without overflowing...
C Relies on a.b mod n = (a mod n)(b mod n) mod n
INTEGER N,X,P !All presumed positive, and X < N.
INTEGER I !A stepper.
INTEGER*8 V,W !Broad scratchpads, otherwise N > 46340 may incur overflow in 32-bit.
V = 1 !=X**0
IF (P.GT.0) THEN !Something to do?
I = P !Yes. Get a copy I can mess with.
W = X !=X**1, X**2, X**4, X**8, ... except, all are mod N.
1 IF (MOD(I,2).EQ.1) V = MOD(V*W,N) !Incorporate W if the low-end calls for it.
I = I/2 !Used. Shift the next one down.
IF (I.GT.0) THEN !Still something to do?
W = MOD(W**2,N) !Yes. Square W ready for the next bit up.
GO TO 1 !Consider it.
END IF !Don't square W if nothing remains. It might overflow.
END IF !Negative powers are ignored.
MODEXP = V !Done, in lb(P) iterations!
END FUNCTION MODEXP !"Bit" presence by arithmetic: works for non-binary arithmetic too.
PROGRAM POKEMR
USE MRTEST
INTEGER I
LOGICAL HIC
DO I = 3,36,2
HIC = MRPRIME(I,6)
WRITE (6,11) I,HIC
11 FORMAT (I6,1X,L)
END DO
END

101
Task/Miller-Rabin-primality-test/FreeBASIC/miller-rabin-primality-test-1.basic
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' version 29-11-2016
' compile with: fbc -s console
' TRUE/FALSE are built-in constants since FreeBASIC 1.04
' But we have to define them for older versions.
#Ifndef TRUE
#Define FALSE 0
#Define TRUE Not FALSE
#EndIf
Function mul_mod(a As ULongInt, b As ULongInt, modulus As ULongInt) As ULongInt
' returns a * b mod modulus
Dim As ULongInt x, y = a ' a mod modulus, but a is already smaller then modulus
While b > 0
If (b And 1) = 1 Then
x = (x + y) Mod modulus
End If
y = (y Shl 1) Mod modulus
b = b Shr 1
Wend
Return x
End Function
Function pow_mod(b As ULongInt, power As ULongInt, modulus As ULongInt) As ULongInt
' returns b ^ power mod modulus
Dim As ULongInt x = 1
While power > 0
If (power And 1) = 1 Then
' x = (x * b) Mod modulus
x = mul_mod(x, b, modulus)
End If
' b = (b * b) Mod modulus
b = mul_mod(b, b, modulus)
power = power Shr 1
Wend
Return x
End Function
Function miller_rabin_test(n As ULongInt, k As Integer) As Byte
If n > 9223372036854775808ull Then ' limit 2^63, pow_mod/mul_mod can't handle bigger numbers
Print "number is to big, program will end"
Sleep
End
End If
' 2 is a prime, if n is smaller then 2 or n is even then n = composite
If n = 2 Then Return TRUE
If (n < 2) OrElse ((n And 1) = 0) Then Return FALSE
Dim As ULongInt a, x, n_one = n - 1, d = n_one
Dim As UInteger s
While (d And 1) = 0
d = d Shr 1
s = s + 1
Wend
While k > 0
k = k - 1
a = Int(Rnd * (n -2)) +2 ' 2 <= a < n
x = pow_mod(a, d, n)
If (x = 1) Or (x = n_one) Then Continue While
For r As Integer = 1 To s -1
x = pow_mod(x, 2, n)
If x = 1 Then Return FALSE
If x = n_one Then Continue While
Next
If x <> n_one Then Return FALSE
Wend
Return TRUE
End Function
' ------=< MAIN >=------
Randomize Timer
Dim As Integer total
Dim As ULongInt y, limit = 2^63-1
For y = limit - 1000 To limit
If miller_rabin_test(y, 5) = TRUE Then
total = total + 1
Print y,
End If
Next
Print : Print
Print total; " primes between "; limit - 1000; " and "; y -1
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End

115
Task/Miller-Rabin-primality-test/FreeBASIC/miller-rabin-primality-test-2.basic
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' version 05-04-2017
' compile with: fbc -s console
' TRUE/FALSE are built-in constants since FreeBASIC 1.04
' But we have to define them for older versions.
#Ifndef TRUE
#Define FALSE 0
#Define TRUE Not FALSE
#EndIf
#Include Once "gmp.bi"
#Macro big_int(a)
Dim As Mpz_ptr a = Allocate( Len( __mpz_struct))
Mpz_init(a)
#EndMacro
Dim Shared As __gmp_randstate_struct rnd_
Function miller_rabin(big_n As Mpz_ptr, num_of_tests As ULong) As Byte
If mpz_cmp_ui(big_n, 1) < 1 Then
Print "Numbers smaller then 1 not allowed"
Sleep 5000
End If
If mpz_cmp_ui(big_n, 2) = 0 OrElse mpz_cmp_ui(big_n, 3) = 0 Then
Return TRUE ' 2 = prime , 3 = prime
End If
If mpz_tstbit(big_n, 0) = 0 Then Return FALSE ' even number, no prime
Dim As ULong r, s
Dim As Byte return_value = TRUE
big_int(n_1) : big_int(n_2) : big_int(a) : big_int(d) : big_int(x)
mpz_sub_ui(n_1, big_n, 1) : mpz_sub_ui(n_2, big_n, 2) : mpz_set(d, n_1)
While mpz_tstbit(d, 0) = 0
mpz_fdiv_q_2exp(d, d, 1)
s += 1
Wend
While num_of_tests > 0
num_of_tests -= 1
mpz_urandomm(a, @rnd_, n_2)
mpz_add_ui(a, a, 2)
mpz_powm(x, a, d, big_n)
If mpz_cmp_ui(x, 1) = 0 Or mpz_cmp(x, n_1) = 0 Then Continue While
For r = 1 To s -1
mpz_powm_ui(x, x, 2, big_n)
If mpz_cmp_ui(x, 1) = 0 Then
return_value = FALSE
Exit While
End If
If mpz_cmp(x, n_1) = 0 Then Continue While
Next
If mpz_cmp(x, n_1) <> 0 Then
Return_value = FALSE
Exit while
End If
Wend
mpz_clear(n_1) : mpz_clear(a) : mpz_clear(d)
mpz_clear(n_2) : mpz_clear(x)
Return return_value
End Function
' ------=< MAIN >=------
Dim As Long x
Dim As String tmp
Dim As ZString Ptr gmp_str : gmp_str = Allocate(1000000)
big_int(big_n)
Randomize Timer
gmp_randinit_mt(@rnd_)
For x = 0 To 200 'create seed for random generator
tmp += Str(Int(Rnd * 10))
Next
Mpz_set_str(big_n, tmp, 10)
gmp_randseed(@rnd_, big_n) ' seed the random number generator
For x = 2 To 100
mpz_set_ui(big_n, x)
If miller_rabin(big_n, 5) = TRUE Then
Print Using "####"; x;
End If
Next
Print : Print
For x = 2 To 3300
mpz_set_ui(big_n, 1)
mpz_mul_2exp(big_n, big_n, x)
mpz_sub_ui(big_n, big_n, 1)
If miller_rabin(big_n, 5) = TRUE Then
gmp_str = Mpz_get_str(0, 10, big_n)
Print "2^";Str(x);"-1 = prime"
End If
Next
gmp_randclear(@rnd_)
mpz_clear(big_n)
DeAllocate(gmp_str)
' empty keyboard buffer
Print : While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End

30
Task/Miller-Rabin-primality-test/FunL/miller-rabin-primality-test.funl
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@ -1,30 +0,0 @@
import util.rnd
def isProbablyPrimeMillerRabin( n, k ) =
d = n - 1
s = 0
while 2|d
s++
d /= 2
repeat k
a = rnd( 2, n )
x = a^d mod n
if x == 1 or x == n - 1 then continue
repeat s - 1
x = x^2 mod n
if x == 1 then return false
if x == n - 1 then break
else
return false
true
for i <- 3..100
if isProbablyPrimeMillerRabin( i, 5 )
println( i )

66
Task/Miller-Rabin-primality-test/Go/miller-rabin-primality-test.go
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@ -1,66 +0,0 @@
package main
import "log"
func main() {
// max uint32 is not prime
c := uint32(1<<32 - 1)
// a few primes near the top of the range. source: prime pages.
for _, p := range []uint32{1<<32 - 5, 1<<32 - 17, 1<<32 - 65, 1<<32 - 99} {
for ; c > p; c-- {
if prime(c) {
log.Fatalf("prime(%d) returned true", c)
}
}
if !prime(p) {
log.Fatalf("prime(%d) returned false", p)
}
c--
}
}
func prime(n uint32) bool {
// bases of 2, 7, 61 are sufficient to cover 2^32
switch n {
case 0, 1:
return false
case 2, 7, 61:
return true
}
// compute s, d where 2^s * d = n-1
nm1 := n - 1
d := nm1
s := 0
for d&1 == 0 {
d >>= 1
s++
}
n64 := uint64(n)
for _, a := range []uint32{2, 7, 61} {
// compute x := a^d % n
x := uint64(1)
p := uint64(a)
for dr := d; dr > 0; dr >>= 1 {
if dr&1 != 0 {
x = x * p % n64
}
p = p * p % n64
}
if x == 1 || uint32(x) == nm1 {
continue
}
for r := 1; ; r++ {
if r >= s {
return false
}
x = x * x % n64
if x == 1 {
return false
}
if uint32(x) == nm1 {
break
}
}
}
return true
}

44
Task/Miller-Rabin-primality-test/Haskell/miller-rabin-primality-test-1.hs
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module Primes where
import System.Random
import System.IO.Unsafe
-- Miller-Rabin wrapped up as an (almost deterministic) pure function
isPrime :: Integer -> Bool
isPrime n = unsafePerformIO (isMillerRabinPrime 100 n)
isMillerRabinPrime :: Int -> Integer -> IO Bool
isMillerRabinPrime k n
| even n = return (n==2)
| n < 100 = return (n `elem` primesTo100)
| otherwise = do ws <- witnesses k n
return $ and [test n (pred n) evens (head odds) a | a <- ws]
where
(evens,odds) = span even (iterate (`div` 2) (pred n))
test :: Integral nat => nat -> nat -> [nat] -> nat -> nat -> Bool
test n n_1 evens d a = x `elem` [1,n_1] || n_1 `elem` powers
where
x = powerMod n a d
powers = map (powerMod n a) evens
witnesses :: (Num a, Ord a, Random a) => Int -> a -> IO [a]
witnesses k n
| n < 9080191 = return [31,73]
| n < 4759123141 = return [2,7,61]
| n < 3474749660383 = return [2,3,5,7,11,13]
| n < 341550071728321 = return [2,3,5,7,11,13,17]
| otherwise = do g <- newStdGen
return $ take k (randomRs (2,n-1) g)
primesTo100 :: [Integer]
primesTo100 = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
-- powerMod m x n = x^n `mod` m
powerMod :: Integral nat => nat -> nat -> nat -> nat
powerMod m x n = f (n - 1) x x `rem` m
where
f d a y = if d==0 then y else g d a y
g i b y | even i = g (i `quot` 2) (b*b `rem` m) y
| otherwise = f (i-1) b (b*y `rem` m)

55
Task/Miller-Rabin-primality-test/Haskell/miller-rabin-primality-test-2.hs
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@ -1,55 +0,0 @@
import Control.Monad (liftM)
import Data.Bits (Bits, testBit, shiftR)
import System.Random (Random, getStdGen, randomRs)
import System.IO.Unsafe (unsafePerformIO)
import Prelude hiding (even, odd)
odd :: (Integral a, Bits a) => a -> Bool
odd = (`testBit` 0)
even :: (Integral a, Bits a) => a -> Bool
even = not . odd
-- modPow - Recursive modular exponentiation by taking successive powers of two
modPow :: (Integral a, Bits a) => a -> a -> a -> a
modPow _ 0 _ = 1
modPow base ex m = let term
| testBit ex 0 = base `mod` m
| otherwise = 1
in (term * modPow (base^2 `mod` m) (ex `shiftR` 1) m) `mod` m
isPrime :: (Integral a, Bits a, Random a) => a -> a -> Bool
isPrime n k
| n < 4 = if n > 1 then True else False -- Deal with 0-3.
| even n = False
| otherwise = let randPool = unsafePerformIO $ randNums (n - 2)
in witness k randPool
where
randNums upper = do
g <- getStdGen
return (randomRs (2, upper) g)
(d, r) = let decompose d r
| odd d = (d, r)
| otherwise = decompose (d `shiftR` 1) (r + 1)
in decompose (n - 1) 0
witness 0 _ = True
witness k (a:rands)
| x == 1 || x == n - 1 = witness (k - 1) rands
| otherwise = check x (r - 1)
where
x = modPow a d n
check _ 0 = False
check x count
| x' == 1 = False
| x' == n - 1 = witness (k - 1) rands
| otherwise = check x' (count - 1)
where x' = modPow x 2 n
-- main function for testing
main :: IO()
main = do
[n,k] <- liftM (map (\x -> read x :: Integer) . words) getLine
print $ isPrime n k

15
Task/Miller-Rabin-primality-test/Haskell/miller-rabin-primality-test-3.hs
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mrt 2 = True
mrt x = odd x && all (\a -> work s x (expMod a d x) ) [2..(x-2)] where
(s,d) = sd (x-1)
sd n = if odd n then (0,n) else (\(x,y) -> (x+1,y)) (sd (n `div` 2))
work 0 n x = (x == 1) || (x == n-1)
work s n x
|(f == 1) && (x /= 1) && (x /= n-1) = False
|otherwise = work (s-1) n f where f = (x*x) `mod` n
expMod a 0 m = 1
expMod a d m
| even d = expMod ((a * a) `mod` m) (d `div` 2) m
| otherwise = (a * expMod a (d - 1) m) `mod` m

33
Task/Miller-Rabin-primality-test/Icon/miller-rabin-primality-test.icon
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@ -1,33 +0,0 @@
procedure main(A)
every n := !A do write(n," is ",(mrp(n,5),"probably prime")|"composite")
end
procedure mrp(n, k)
if n = 2 then return ""
if n%2 = 0 then fail
nm1 := decompose(n-1)
s := nm1[1]
d := nm1[2]
every !k do {
a := ?(n-2)+1
x := (a^d)%n
if x = (1|(n-1)) then next
every !(s-1) do {
x := (x*x)%n
if x = 1 then fail
if x = (n-1) then break next
}
fail
}
return ""
end
procedure decompose(nm1)
s := 1
d := nm1
while d%2 = 0 do {
d /:= 2
s +:= 1
}
return [s,d]
end

9
Task/Miller-Rabin-primality-test/Java/miller-rabin-primality-test-1.java
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@ -1,9 +0,0 @@
import java.math.BigInteger;
public class MillerRabinPrimalityTest {
public static void main(String[] args) {
BigInteger n = new BigInteger(args[0]);
int certainty = Integer.parseInt(args[1]);
System.out.println(n.toString() + " is " + (n.isProbablePrime(certainty) ? "probably prime" : "composite"));
}
}

79
Task/Miller-Rabin-primality-test/Java/miller-rabin-primality-test-2.java
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import java.math.BigInteger;
public class Prime {
// this is the RabinMiller test, deterministically correct for n < 341,550,071,728,321
// http://rosettacode.org/wiki/Miller-Rabin_primality_test#Python:_Proved_correct_up_to_large_N
public static boolean isPrime(BigInteger n, int precision) {
if (n.compareTo(new BigInteger("341550071728321")) >= 0) {
return n.isProbablePrime(precision);
}
int intN = n.intValue();
if (intN == 1 || intN == 4 || intN == 6 || intN == 8) return false;
if (intN == 2 || intN == 3 || intN == 5 || intN == 7) return true;
int[] primesToTest = getPrimesToTest(n);
if (n.equals(new BigInteger("3215031751"))) {
return false;
}
BigInteger d = n.subtract(BigInteger.ONE);
BigInteger s = BigInteger.ZERO;
while (d.mod(BigInteger.valueOf(2)).equals(BigInteger.ZERO)) {
d = d.shiftRight(1);
s = s.add(BigInteger.ONE);
}
for (int a : primesToTest) {
if (try_composite(a, d, n, s)) {
return false;
}
}
return true;
}
public static boolean isPrime(BigInteger n) {
return isPrime(n, 100);
}
public static boolean isPrime(int n) {
return isPrime(BigInteger.valueOf(n), 100);
}
public static boolean isPrime(long n) {
return isPrime(BigInteger.valueOf(n), 100);
}
private static int[] getPrimesToTest(BigInteger n) {
if (n.compareTo(new BigInteger("3474749660383")) >= 0) {
return new int[]{2, 3, 5, 7, 11, 13, 17};
}
if (n.compareTo(new BigInteger("2152302898747")) >= 0) {
return new int[]{2, 3, 5, 7, 11, 13};
}
if (n.compareTo(new BigInteger("118670087467")) >= 0) {
return new int[]{2, 3, 5, 7, 11};
}
if (n.compareTo(new BigInteger("25326001")) >= 0) {
return new int[]{2, 3, 5, 7};
}
if (n.compareTo(new BigInteger("1373653")) >= 0) {
return new int[]{2, 3, 5};
}
return new int[]{2, 3};
}
private static boolean try_composite(int a, BigInteger d, BigInteger n, BigInteger s) {
BigInteger aB = BigInteger.valueOf(a);
if (aB.modPow(d, n).equals(BigInteger.ONE)) {
return false;
}
for (int i = 0; BigInteger.valueOf(i).compareTo(s) < 0; i++) {
// if pow(a, 2**i * d, n) == n-1
if (aB.modPow(BigInteger.valueOf(2).pow(i).multiply(d), n).equals(n.subtract(BigInteger.ONE))) {
return false;
}
}
return true;
}
}

24
Task/Miller-Rabin-primality-test/JavaScript/miller-rabin-primality-test.js
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@ -1,24 +0,0 @@
function probablyPrime(n) {
if (n === 2 || n === 3) return true
if (n % 2 === 0 || n < 2) return false
// Write (n - 1) as 2^s * d
var s = 0,
d = n - 1
while ((d & 1) == 0) {
d >>= 1
++s
}
let base = 2
var x = Math.pow(base, d) % n
if (x == 1 || x == n - 1) return true
for (var i = 1; i <= s; i++) {
x = (x * x) % n
if (x === n - 1) return true
}
return false
}

27
Task/Miller-Rabin-primality-test/Julia/miller-rabin-primality-test.jl
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@ -1,27 +0,0 @@
witnesses(n::Union(Uint8,Int8,Uint16,Int16)) = (2,3)
witnesses(n::Union(Uint32,Int32)) = n < 1373653 ? (2,3) : (2,7,61)
witnesses(n::Union(Uint64,Int64)) =
n < 1373653 ? (2,3) :
n < 4759123141 ? (2,7,61) :
n < 2152302898747 ? (2,3,5,7,11) :
n < 3474749660383 ? (2,3,5,7,11,13) :
(2,325,9375,28178,450775,9780504,1795265022)
function isprime(n::Integer)
n == 2 && return true
(n < 2) | iseven(n) && return false
s = trailing_zeros(n-1)
d = (n-1) >>> s
for a in witnesses(n)
a < n || break
x = powermod(a,d,n)
x == 1 && continue
t = s
while x != n-1
(t-=1) <= 0 && return false
x = oftype(n, Base.widemul(x,x) % n)
x == 1 && return false
end
end
return true
end

53
Task/Miller-Rabin-primality-test/Kotlin/miller-rabin-primality-test.kts
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@ -1,53 +0,0 @@
// version 1.1.2
import java.math.BigInteger
import java.util.Random
val bigTwo = BigInteger.valueOf(2L)
fun isProbablyPrime(n: BigInteger, k: Int): Boolean {
require (n > bigTwo && n % bigTwo == BigInteger.ONE) { "Must be odd and greater than 2" }
var s = 0
val nn = n - BigInteger.ONE
var d: BigInteger
do {
s++
d = nn.shiftRight(s)
}
while (d % bigTwo == BigInteger.ZERO)
val rand = Random()
loop@ for (i in 1..k) {
var a: BigInteger
do {
a = BigInteger(n.bitLength(), rand)
}
while(a < bigTwo || a > nn) // make sure it's in the interval [2, n - 1]
var x = a.modPow(d, n)
if (x == BigInteger.ONE || x == nn) continue
for (r in 1 until s) {
x = (x * x) % n
if (x == BigInteger.ONE) return false
if (x == nn) break@loop
}
return false
}
return true
}
fun main(args: Array<String>) {
val k = 20 // say
// obtain all primes up to 100
println("The following numbers less than 100 are prime:")
for (i in 3..99 step 2)
if (isProbablyPrime(BigInteger.valueOf(i.toLong()), k)) print("$i ")
println("\n")
// check if some big numbers are probably prime
val bia = arrayOf(
BigInteger("4547337172376300111955330758342147474062293202868155909489"),
BigInteger("4547337172376300111955330758342147474062293202868155909393")
)
for (bi in bia)
println("$bi is ${if (isProbablyPrime(bi, k)) "probably prime" else "composite"}")
}

290
Task/Miller-Rabin-primality-test/Liberty-BASIC/miller-rabin-primality-test.basic
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@ -1,290 +0,0 @@
DIM mersenne(11)
mersenne(1)=7
mersenne(2)=31
mersenne(3)=127
mersenne(4)=8191
mersenne(5)=131071
mersenne(6)=524287
mersenne(7)=2147483647
mersenne(8)=2305843009213693951
mersenne(9)=618970019642690137449562111
mersenne(10)=162259276829213363391578010288127
mersenne(11)=170141183460469231731687303715884105727
dim SmallPrimes(1000)
data 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
data 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
data 73, 79, 83, 89, 97, 101, 103, 107, 109, 113
data 127, 131, 137, 139, 149, 151, 157, 163, 167, 173
data 179, 181, 191, 193, 197, 199, 211, 223, 227, 229
data 233, 239, 241, 251, 257, 263, 269, 271, 277, 281
data 283, 293, 307, 311, 313, 317, 331, 337, 347, 349
data 353, 359, 367, 373, 379, 383, 389, 397, 401, 409
data 419, 421, 431, 433, 439, 443, 449, 457, 461, 463
data 467, 479, 487, 491, 499, 503, 509, 521, 523, 541
data 547, 557, 563, 569, 571, 577, 587, 593, 599, 601
data 607, 613, 617, 619, 631, 641, 643, 647, 653, 659
data 661, 673, 677, 683, 691, 701, 709, 719, 727, 733
data 739, 743, 751, 757, 761, 769, 773, 787, 797, 809
data 811, 821, 823, 827, 829, 839, 853, 857, 859, 863
data 877, 881, 883, 887, 907, 911, 919, 929, 937, 941
data 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013
data 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069
data 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151
data 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223
data 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291
data 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373
data 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451
data 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511
data 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583
data 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657
data 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733
data 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811
data 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889
data 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987
data 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053
data 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129
data 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213
data 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287
data 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357
data 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423
data 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531
data 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617
data 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687
data 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741
data 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819
data 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903
data 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999
data 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079
data 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181
data 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257
data 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331
data 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413
data 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511
data 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571
data 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643
data 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727
data 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821
data 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907
data 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989
data 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057
data 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139
data 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231
data 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297
data 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409
data 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493
data 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583
data 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657
data 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751
data 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831
data 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937
data 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003
data 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087
data 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179
data 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279
data 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387
data 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443
data 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521
data 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639
data 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693
data 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791
data 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857
data 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939
data 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053
data 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133
data 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221
data 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301
data 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367
data 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473
data 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571
data 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673
data 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761
data 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833
data 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917
data 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997
data 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103
data 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207
data 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297
data 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411
data 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499
data 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561
data 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643
data 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723
data 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829
data 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919
print "Liberty Miller Rabin Demonstration"
print "Loading Small Primes"
for i=1 to 1000: read x : SmallPrimes(i)=x :next :NoOfSmallPrimes=1000
print NoOfSmallPrimes;" Primes Loaded"
'Prompt "Enter number to test:";resp$
'x=val(resp$)
'goto [Jump]
For i=1 to 11
x=mersenne(i)
t1=time$("ms")
[TryAnother]
print
iterations=0
[Loop]
iterations=iterations+1
if MillerRabin(x,7)=1 then
t2=time$("ms")
print "Composite, found in ";t2-t1;" milliseconds"
else
t2=time$("ms")
print x;" Probably Prime. Tested in ";t2-t1;" milliseconds"
playwave "tada.wav", async
end if
print
next
END
Function GCD( m,n )
' Find greatest common divisor with Extend Euclidian Algorithm
' Knuth Vol 1 P.13 Algorithm E
ap =1 :b =1 :a =0 :bp =0: c =m :d =n
[StepE2]
q = int(c/d) :r = c-q*d
if r<>0 then
c=d :d=r :t=ap :ap=a :a=t-q*a :t=bp :bp=b :b=t-q*b
'print ap;" ";b;" ";a;" ";bp;" ";c;" ";d;" ";t;" ";q
goto [StepE2]
end if
GCD=a*m+b*n
'print ap;" ";b;" ";a;" ";bp;" ";c;" ";d;" ";t;" ";q
End Function 'Extended Euclidian GCD
function IsEven( x )
if ( x MOD 2 )=0 then
IsEven=1
else
IsEven=0
end if
end function
function IsOdd( x )
if ( x MOD 2 )=0 then
IsOdd=0
else
IsOdd=1
end if
end function
Function FastExp(x, y, N)
if (y=1) then 'MOD(x,N)
FastExp=x-int(x/N)*N
goto [ExitFunction]
end if
if ( y and 1) = 0 then
dum1=y/2
dum2=y-int(y/2)*2 'MOD(y,2)
temp=FastExp(x,dum1,N)
z=temp*temp
FastExp=z-int(z/N)*N 'MOD(temp*temp,N)
goto [ExitFunction]
else
dum1=y-1
dum1=dum1/2
temp=FastExp(x,dum1,N)
dum2=temp*temp
temp=dum2-int(dum2/N)*N 'MOD(dum2,N)
z=temp*x
FastExp=z-int(z/N)*N 'MOD(temp*x,N)
goto [ExitFunction]
end if
[ExitFunction]
end function
Function MillerRabin(n,b)
'print "Miller Rabin"
't1=time$("ms")
if IsEven(n) then
MillerRabin=1
goto [ExtFn]
end if
i=0
[Loop]
i=i+1
if i>1000 then goto [Continue]
if ( n MOD SmallPrimes(i) )=0 then
MillerRabin=0
goto [ExtFn]
end if
goto [Loop]
[Continue]
if GCD(n,b)>1 then
MillerRabin=1
goto [ExtFn]
end if
q=n-1
t=0
while (int(q) AND 1 )=0
t=t+1
q=int(q/2)
wend
r=FastExp(b, q, n)
if ( r <> 1 ) then
e=0
while ( e < (t-1) )
if ( r <> (n-1) ) then
r=FastExp(r, r, n)
else
Exit While
end if
e=e+1
wend
[ExitLoop]
end if
if ( (r=1) OR (r=(n-1)) ) then
MillerRabin=0
else
MillerRabin=1
end if
[ExtFn]
End Function

59
Task/Miller-Rabin-primality-test/Lua/miller-rabin-primality-test.lua
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@ -1,59 +0,0 @@
function MRIsPrime(n, k)
-- If n is prime, returns true (without fail).
-- If n is not prime, then returns false with probability ≥ 4^(-k), true otherwise.
-- First, deal with 1 and multiples of 2.
if n == 1 then
return false
elseif n == 2 then
return true
elseif n%2 == 0 then
return false
end
-- Now n is odd and greater than 1.
-- Find the unique expression n = 1 + t*2^h for n, where t is odd and h≥1.
t = (n-1)/2
h = 1
while t%2 == 0 do
t = t/2
h = h + 1
end
for i = 1, k do
-- Generate a random integer between 1 and n-1 inclusive.
a = math.random(n-1)
-- Test whether a is an element of the set L, and return false if not.
if not IsInL(n, a, t, h) then
return false
end
end
-- All generated a were in the set L; thus with high probability, n is prime.
return true
end
function IsInL(n, a, t, h)
local b = PowerMod(a, t, n)
if b == 1 then
return true
end
for j = 0, h-1 do
if b == n-1 then
return true
elseif b == 1 then
return false
end
b = (b^2)%n
end
return false
end
function PowerMod(x, y, m)
-- Computes x^y mod m.
local z = 1
while y > 0 do
if y%2 == 0 then
x, y, z = (x^2)%m, y//2, z
else
x, y, z = (x^2)%m, y//2, (x*z)%m
end
end
return z
end

59
Task/Miller-Rabin-primality-test/M2000-Interpreter/miller-rabin-primality-test.m2000
View file

@ -1,59 +0,0 @@
function isProbablyPrime(n as *BigInteger, k as long) {
boolean T=true, F=false
=F
Zero=BigInteger("0")
One=BigInteger("1")
Two=BigInteger("2")
Method n, "compare", Two as c1
Method n, "modulus", Two as m2
method m2,"compare", zero as C2
if c1=0 or c2=0 then exit
with n, "toString" as ns$
s=0
Method n, "subtract", one as nn
d=nn
with d, "tostring" as dstr$
do
method d,"modulus", two as m2
method m2,"compare", zero as C
if c else exit
s++
method d, "divide", two as d
Always
z=len(ns$)
a=one
with a, "toString" as astr$
x=a
=T
for i=1 to k {
do
zs=""
for j=1 to len(ns$)
zs+=chr$(47+random(1,10))
next
a=bigInteger(zs)
method nn,"compare", a as C
method a,"compare", one as c1
until c=1 and c1>-1
method a, "modpow", d, n as x
method x,"compare", one as c1
if c1 else continue
method x,"compare", nn as c1
if c1 else continue
for r=1 to s {
method x, "modpow", two, n as x
method x,"compare", one as c1
if c1 else =F : break
method x,"compare", nn as c1
if c1 else exit
}
if c1 then =F: break
}
}
profiler
Print isProbablyPrime(BigInteger("5400349"), 5)=True
print timecount
profiler
a=BigInteger("5400349"): Method a, "isProbablyPrime", 5 as ret:Print ret
print timecount

9
Task/Miller-Rabin-primality-test/Mathematica/miller-rabin-primality-test-1.math
View file

@ -1,9 +0,0 @@
MillerRabin[n_,k_]:=Module[{d=n-1,s=0,test=True},While[Mod[d,2]==0 ,d/=2 ;s++]
Do[
a=RandomInteger[{2,n-1}]; x=PowerMod[a,d,n];
If[x!=1,
For[ r = 0, r < s, r++, If[x==n-1, Continue[]]; x = Mod[x*x, n]; ];
If[ x != n-1, test=False ];
];
,{k}];
Print[test] ]

2
Task/Miller-Rabin-primality-test/Mathematica/miller-rabin-primality-test-2.math
View file

@ -1,2 +0,0 @@
MillerRabin[17388,10]
->False

47
Task/Miller-Rabin-primality-test/Maxima/miller-rabin-primality-test.maxima
View file

@ -1,47 +0,0 @@
/* Miller-Rabin algorithm is builtin, see function primep. Here is another implementation */
/* find highest power of p, p^s, that divide n, and return s and n / p^s */
facpow(n, p) := block(
[s: 0],
while mod(n, p) = 0 do (s: s + 1, n: quotient(n, p)),
[s, n]
)$
/* check whether n is a strong pseudoprime to base a; s and d are given by facpow(n - 1, 2) */
sppp(n, a, s, d) := block(
[x: power_mod(a, d, n), q: false],
if x = 1 or x = n - 1 then true else (
from 2 thru s do (
x: mod(x * x, n),
if x = 1 then return(q: false) elseif x = n - 1 then return(q: true)
),
q
)
)$
/* Miller-Rabin primality test. For n < 341550071728321, the test is deterministic;
for larger n, the number of bases tested is given by the option variable
primep_number_of_tests, which is used by Maxima in primep. The bound for deterministic
test is also the same as in primep. */
miller_rabin(n) := block(
[v: [2, 3, 5, 7, 11, 13, 17], s, d, q: true, a],
if n < 19 then member(n, v) else (
[s, d]: facpow(n - 1, 2),
if n < 341550071728321 then ( /* see http://oeis.org/A014233 */
for a in v do (
if not sppp(n, a, s, d) then return(q: false)
),
q
) else (
thru primep_number_of_tests do (
a: 2 + random(n - 3),
if not sppp(n, a, s, d) then return(q: false)
),
q
)
)
)$

218
Task/Miller-Rabin-primality-test/Mercury/miller-rabin-primality-test.mercury
View file

@ -1,218 +0,0 @@
%----------------------------------------------------------------------%
:- module primality.
:- interface.
:- import_module integer.
:- pred is_prime(integer::in, integer::out) is multi.
%----------------------------------------------------------------------%
:- implementation.
:- import_module bool, int, list, math, require, string.
%----------------------------------------------------------------------%
% is_prime/2 implements a Miller-Rabin primality test, is
% deterministic for N < 3.415e+14, and is probabilistic for
% larger N. Returns integer(0) if not prime, integer(1) if prime,
% and -integer(1) if fails.
% :- pred is_prime(integer::in, integer::out) is multi.
is_prime(N, P) :-
N < integer(2), P = integer(0).
is_prime(N, P) :-
N = integer(2), P = integer(1).
is_prime(N, P) :-
N = integer(3), P = integer(1).
is_prime(N, P) :- %% even numbers > 3: false
N > integer(3),
(N mod integer(2)) = integer(0),
P = integer(0).
%%-------------deterministic--------
is_prime(N, P) :- %% 3 < odd number < 3.415e+14
N > integer(3),
(N mod integer(2)) = integer(1),
N < integer(341550071728321),
deterministic_witnesses(N, DList),
is_mr_prime(N, DList, R),
P = R.
%%-------------probabilistic--------
is_prime(N, P) :- %% 3.415e+14 =< odd number
N > integer(3),
(N mod integer(2)) = integer(1),
N >= integer(341550071728321),
random_witnesses(N, 20, RList),
is_mr_prime(N, RList, R),
P = R.
is_prime(_N, P) :- P = -integer(1).
%----------------------------------------------------------------------%
% returns list of deterministic witnesses
:- pred deterministic_witnesses(integer::in,
list(integer)::out) is multi.
deterministic_witnesses(N, L) :- N < integer(1373653),
L = [integer(2), integer(3)].
deterministic_witnesses(N, L) :- N < integer(9080191),
L = [integer(31), integer(73)].
deterministic_witnesses(N, L) :- N < integer(25326001),
L = [integer(2), integer(3), integer(5)].
deterministic_witnesses(N, L) :- N < integer(3215031751),
L = [integer(2), integer(3), integer(5), integer(7)].
deterministic_witnesses(N, L) :- N < integer(4759123141),
L = [integer(2), integer(7), integer(61)].
deterministic_witnesses(N, L) :- N < integer(1122004669633),
L = [integer(2), integer(13), integer(23), integer(1662803)].
deterministic_witnesses(N, L) :- N < integer(2152302898747),
L = [integer(2), integer(3), integer(5), integer(7), integer(11)].
deterministic_witnesses(N, L) :- N < integer(3474749660383),
L = [integer(2), integer(3), integer(5), integer(7), integer(11),
integer(13)].
deterministic_witnesses(N, L) :- N < integer(341550071728321),
L = [integer(2), integer(3), integer(5), integer(7),
integer(11), integer(13), integer(17)].
deterministic_witnesses(_N, L) :- L = []. %% signals failure
%----------------------------------------------------------------------%
%% random_witnesses/3 receives an integer, X, an int, K, and
%% returns a list, P, of K pseudo-random integers in a range
%% 1 to X-1.
:- pred random_witnesses(integer::in, int::in,
list(integer)::out) is det.
random_witnesses(X, K, P) :-
A = integer(6364136223846793005),
B = integer(1442695040888963407),
C = X - integer(2),
rw_loop(X, A, B, C, K, [], P).
:- pred rw_loop(integer::in, integer::in, integer::in, integer::in,
int::in, list(integer)::in, list(integer)::out) is det.
rw_loop(X, A, B, C, K, L, P) :-
X1 = (((X * A) + B) mod C) + integer(1),
( if K = 0 then P = L
else rw_loop(X1, A, B, C, K-1, [X1|L], P)
).
%----------------------------------------------------------------------%
% is_mr_prime/2 receives integer N and list As and returns true if
% N is probably prime, and false otherwise
:- pred is_mr_prime(integer::in, list(integer)::in, integer::out) is nondet.
is_mr_prime(N, As, R) :-
find_ds(N, L),
L = [D|T],
T = [S|_],
outer_loop(N, As, D, S, R).
:- pred outer_loop(integer::in, list(integer)::in, integer::in,
integer::in, integer::out) is nondet.
outer_loop(N, As, D, S, R) :-
As = [A|At],
Base = powm(A, D, N), %% = A^D mod N
inner_loop(Base, N, integer(0), S, U),
( if U = integer(0) then R = integer(0)
else ( if At = [] then R = integer(1)
else outer_loop(N, At, D, S, R)
)
).
:- pred inner_loop(integer::in, integer::in, integer::in,
integer::in, integer::out) is multi.
inner_loop(Base, N, Loop, S, U) :-
Next_Base = (Base * Base) mod N,
Next_Loop = Loop + integer(1),
( if Loop = integer(0) then
( if Base = integer(1) then U = integer(1) % true
else if Base = N - integer(1) then U = integer(1) % true
else if Next_Loop = S then U = integer(0) % false
else inner_loop(Next_Base, N, Next_Loop, S, U)
)
else if Base = N - integer(1) then U = integer(1) % true
else if Next_Loop = S then U = integer(0) % false
else inner_loop(Next_Base, N, Next_Loop, S, U)
).
%----------------------------------------------------------------------%
% find_ds/2 receives odd integer N
% and returns [D, S] such that N-1 = 2^S * D
:- pred find_ds(integer::in, list(integer)::out) is multi.
find_ds(N, L) :-
A = N - integer(1),
find_ds1(A, integer(0), L).
:- pred find_ds1(integer::in, integer::in, list(integer)::out) is multi.
find_ds1(D, S, L) :-
D mod integer(2) = integer(0),
P = D div integer(2),
Q = S + integer(1),
find_ds1(P, Q, L).
find_ds1(D, S, L) :-
L = [D, S].
%----------------------------------------------------------------------%
:- func powm(integer, integer, integer) = integer.
% computes A^D mod N
powm(A, D, N) =
( if D = integer(0) then integer(1)
else ( if (D mod integer(2)) = integer(0) then
(integer.pow(powm(A, (D div integer(2)), N),
integer(2))) mod N
else (A * powm(A, D - integer(1), N)) mod N
)
).
%----------------------------------------------------------------------%
:- end_module primality.
% A means of testing the predicate is_prime/2
%----------------------------------------------------------------------%
:- module test_is_prime.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is cc_multi.
%----------------------------------------------------------------------%
:- implementation.
:- import_module bool, char, int, integer, list, math, require, string.
:- import_module primality.
%----------------------------------------------------------------------%
% TEST THE IS_PRIME PREDICATE
% $ ./test_is_prime <integer>
%---------------------------------------------%
main(!IO) :-
command_line_arguments(Args, !IO),
filter(is_all_digits, Args, CleanArgs),
Arg1 = list.det_index0(CleanArgs, 0),
M = integer.det_from_string(Arg1),
is_prime(M,P),
io.format(" is_prime(%s) = ", [s(integer.to_string(M))], !IO),
( if P = integer(0) then io.write_string("false.\n", !IO)
else if P = integer(1) then io.write_string("true.\n", !IO)
else if P = -integer(1) then
io.write_string("N fails all tests.\n", !IO)
else io.write_string("Has reported neither true nor false
nor any error condition.\n", !IO)
).
%----------------------------------------------------------------------%
:- end_module test_is_prime.

73
Task/Miller-Rabin-primality-test/Nim/miller-rabin-primality-test-1.nim
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@ -1,73 +0,0 @@
## Nim currently doesn't have a BigInt standard library
## so we translate the version from Go which uses a
## deterministic approach, which is correct for all
## possible values in uint32.
proc isPrime*(n: uint32): bool =
# bases of 2, 7, 61 are sufficient to cover 2^32
case n
of 0, 1: return false
of 2, 7, 61: return true
else: discard
var
nm1 = n-1
d = nm1.int
s = 0
n = n.uint64
while d mod 2 == 0:
d = d shr 1
s += 1
for a in [2, 7, 61]:
var
x = 1.uint64
p = a.uint64
dr = d
while dr > 0:
if dr mod 2 == 1:
x = x * p mod n
p = p * p mod n
dr = dr shr 1
if x == 1 or x.uint32 == nm1:
continue
var r = 1
while true:
if r >= s:
return false
x = x * x mod n
if x == 1:
return false
if x.uint32 == nm1:
break
r += 1
return true
proc isPrime*(n: int32): bool =
## Overload for int32
n >= 0 and n.uint32.isPrime
when isMainModule:
const primeNumber1000 = 7919 # source: https://en.wikipedia.org/wiki/List_of_prime_numbers
var
i = 0u32
numberPrimes = 0
while true:
if isPrime(i):
if numberPrimes == 999:
break
numberPrimes += 1
i += 1
assert i == primeNumber1000
assert isPrime(2u32)
assert isPrime(31u32)
assert isPrime(37u32)
assert isPrime(1123u32)
assert isPrime(492366587u32)
assert isPrime(1645333507u32)

140
Task/Miller-Rabin-primality-test/Nim/miller-rabin-primality-test-2.nim
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# Compile as: $ nim c -d:release mrtest.nim
# Run using: $ ./mrtest
import math # for gcd and mod
import bitops # for countTrailingZeroBits
import strutils, typetraits # for number input
import times, os # for timing code execution
proc addmod*[T: SomeInteger](a, b, modulus: T): T =
## Modular addition
let a_m = if a < modulus: a else: a mod modulus
if b == 0.T: return a_m
let b_m = if b < modulus: b else: b mod modulus
# Avoid doing a + b that could overflow here
let b_from_m = modulus - b_m
if a_m >= b_from_m: return a_m - b_from_m
return a_m + b_m # safe to add here; a + b < modulus
proc mulmod*[T: SomeInteger](a, b, modulus: T): T =
## Modular multiplication
var a_m = if a < modulus: a else: a mod modulus
var b_m = if b < modulus: b else: b mod modulus
if b_m > a_m: swap(a_m, b_m)
while b_m > 0.T:
if (b_m and 1) == 1: result = addmod(result, a_m, modulus)
a_m = (a_m shl 1) - (if a_m >= (modulus - a_m): modulus else: 0)
b_m = b_m shr 1
proc expmod*[T: SomeInteger](base, exponent, modulus: T): T =
## Modular exponentiation
result = 1 # (exp 0 = 1)
var (e, b) = (exponent, base)
while e > 0.T:
if (e and 1) == 1: result = mulmod(result, b, modulus)
e = e shr 1
b = mulmod(b, b, modulus)
# Returns true if +self+ passes Miller-Rabin Test on witnesses +b+
proc miller_rabin_test[T: SomeInteger](num: T, witnesses: seq[uint64]): bool =
var d = num - 1
let (neg_one_mod, n) = (d, d)
d = d shr countTrailingZeroBits(d) # suck out factors of 2 from d
for b in witnesses: # do M-R test with each witness base
if b.T mod num == 0: continue # **skip base if a multiple of input**
var s = d
var y = expmod(b.T, d, num)
while s != n and y != 1 and y != neg_one_mod:
y = mulmod(y, y, num)
s = s shl 1
if y != neg_one_mod and (s and 1) != 1: return false
true
proc selectWitnesses[T: SomeInteger](num: T): seq[uint64] =
## Best known deterministic witnesses for given range and number of bases
## https://miller-rabin.appspot.com/
## https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
if num < 341_531u:
result = @[9345883071009581737u64]
elif num < 1_050_535_501u:
result = @[336781006125u64, 9639812373923155u64]
elif num < 350_269_456_337u:
result = @[4230279247111683200u64, 14694767155120705706u64, 16641139526367750375u64]
elif num < 55_245_642_489_451u:
result = @[2u64, 141889084524735u64, 1199124725622454117u64, 11096072698276303650u64]
elif num < 7_999_252_175_582_851u:
result = @[2u64, 4130806001517u64, 149795463772692060u64, 186635894390467037u64, 3967304179347715805u64]
elif num < 585_226_005_592_931_977u:
result = @[2u64, 123635709730000u64, 9233062284813009u64, 43835965440333360u64, 761179012939631437u64, 1263739024124850375u64]
elif num.uint64 < 18_446_744_073_709_551_615u64:
result = @[2u64, 325, 9375, 28178, 450775, 9780504, 1795265022]
else:
result = @[2u64, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
proc primemr*[T: SomeInteger](n: T): bool =
let primes = @[2u64, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
if n <= primes[^1].T: return (n in primes) # for n <= primes.last
let modp47 = 614889782588491410u # => primes.product, largest < 2^64
if gcd(n, modp47) != 1: return false # eliminates 86.2% of all integers
let witnesses = selectWitnesses(n)
miller_rabin_test(n, witnesses)
echo "\nprimemr?"
echo("n = ", 1645333507u)
var te = epochTime()
echo primemr 1645333507u
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
echo("n = ", 2147483647u)
te = epochTime()
echo primemr 2147483647u
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
echo("n = ", 844674407370955389u)
te = epochTime()
echo primemr 844674407370955389u
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
echo("n = ", 1844674407370954349u)
te = epochTime()
echo primemr 1844674407370954349u
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
echo("n = ", 1844674407370954351u)
te = epochTime()
echo primemr 1844674407370954351u
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
echo("n = ", 9223372036854775783u)
te = epochTime()
echo primemr 9223372036854775783u
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
echo("n = ", 9241386435364257883u64)
te = epochTime()
echo primemr 9241386435364257883u64
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
echo("n = ", 18446744073709551533u64, ", is largest prime < 2^64")
te = epochTime()
echo 18446744073709551533u64.primemr
echo (epochTime()-te).formatFloat(ffDecimal, 6)
echo "\nprimemr?"
let num = 5_000_000u # => 348_513 primes
var primes: seq[uint] = @[]
echo("find primes < ", num)
te = epochTime()
for n in 0u..num:
if n.primemr: primes.add(n)
stdout.write("\r",((float64(n) / float64(num))*100).formatFloat(ffDecimal, 1), "%")
echo("\nnumber of primes < ",num, " are ", primes.len)
echo (epochTime()-te).formatFloat(ffDecimal, 6)

41
Task/Miller-Rabin-primality-test/Nim/miller-rabin-primality-test-3.nim
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@ -1,41 +0,0 @@
(* Translated from the wikipedia pseudo-code *)
let miller_rabin n ~iter:k =
(* return r and d where n = 2^r*d (from scheme implementation) *)
let get_rd n =
let rec loop r d =
(* not even *)
if Z.(equal (logand d one) one) then
(r,d)
else
loop Z.(r + one) Z.(div d ~$2)
in
loop Z.zero n
in
let single_miller n r d =
(* (random (n - 4)) + 2 *)
let a = Bigint.to_zarith_bigint
Bigint.((random ((of_zarith_bigint n) - (of_int 4))) + (of_int 2))
in
let x = Z.(powm a d n) in
if Z.(equal x ~$1) || Z.(equal x (n - ~$1)) then true
else
let rec loop i x =
if Z.(equal ~$i (r - ~$1)) then false
else
let x = Z.(powm x ~$2 n) in
if Z.(equal x (n - ~$1)) then true
else loop (i + 1) x
in
loop 0 x
in
let n = Z.abs n in
if Z.(equal n one) then false
else if Z.(equal (logand n one) zero) then false
else if Z.(equal (n mod ~$3) zero) then false
else
let r, d = get_rd Z.(n - one) in
let rec loop i bool =
if i = k then bool
else loop (i + 1) (bool && single_miller n r d)
in
loop 0 true

104
Task/Miller-Rabin-primality-test/Oz/miller-rabin-primality-test.oz
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@ -1,104 +0,0 @@
%--------------------------------------------------------------------------%
% module: Primality
% file: Primality.oz
% version: 17 DEC 2014 @ 6:50AM
%--------------------------------------------------------------------------%
declare
%--------------------------------------------------------------------------%
fun {IsPrime N} % main interface of module
if N < 2 then false
elseif N < 4 then true
elseif (N mod 2) == 0 then false
elseif N < 341330071728321 then {IsMRprime N {DetWit N}}
else {IsMRprime N {ProbWit N 20}}
end
end
%--------------------------------------------------------------------------%
fun {DetWit N} % deterministic witnesses
if N < 1373653 then [2 3]
elseif N < 9080191 then [31 73]
elseif N < 25326001 then [2 3 5]
elseif N < 3215031751 then [2 3 5 7]
elseif N < 4759123141 then [2 7 61]
elseif N < 1122004669633 then [2 13 23 1662803]
elseif N < 2152302898747 then [2 3 5 7 11]
elseif N < 3474749660383 then [2 3 5 7 11 13]
elseif N < 341550071728321 then [2 3 5 7 11 13 17]
else nil
end
end
%--------------------------------------------------------------------------%
fun {ProbWit N K} % probabilistic witnesses
local A B C in
A = 6364136223846793005
B = 1442695040888963407
C = N - 2
{RWloop N A B C K nil}
end
end
fun {RWloop N A B C K L}
local N1 in
N1 = (((N * A) + B) mod C) + 1
if K == 0 then L
else {RWloop N1 A B C (K - 1) N1|L}
end
end
end
%--------------------------------------------------------------------------%
fun {IsMRprime N As} % the Miller-Rabin algorithm
local D S T Ts in
{FindDS N} = D|S
{OuterLoop N As D S}
end
end
fun {OuterLoop N As D S}
local A At Base C in
As = A|At
Base = {Powm A D N}
C = {InnerLoop Base N 0 S}
if {Not C} then false
elseif {And C (At == nil)} then true
else {OuterLoop N At D S}
end
end
end
fun {InnerLoop Base N Loop S}
local NextBase NextLoop in
NextBase = (Base * Base) mod N
NextLoop = Loop + 1
if {And (Loop == 0) (Base == 1)} then true
elseif Base == (N - 1) then true
elseif NextLoop == S then false
else {InnerLoop NextBase N NextLoop S}
end
end
end
%--------------------------------------------------------------------------%
fun {FindDS N}
{FindDS1 (N - 1) 0}
end
fun {FindDS1 D S}
if (D mod 2 == 0) then {FindDS1 (D div 2) (S + 1)}
else D|S
end
end
%--------------------------------------------------------------------------%
fun {Powm A D N} % returns (A ^ D) mod N
if D == 0 then 1
elseif (D mod 2) == 0 then {Pow {Powm A (D div 2) N} 2} mod N
else (A * {Powm A (D - 1) N}) mod N
end
end
%--------------------------------------------------------------------------%
% end_module Primality

1
Task/Miller-Rabin-primality-test/PARI-GP/miller-rabin-primality-test-1.parigp
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@ -1 +0,0 @@
MR(n,k)=ispseudoprime(n,k);

16
Task/Miller-Rabin-primality-test/PARI-GP/miller-rabin-primality-test-2.parigp
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@ -1,16 +0,0 @@
sprp(n,b)={
my(s = valuation(n-1, 2), d = Mod(b, n)^(n >> s));
if (d == 1, return(1));
for(i=1,s-1,
if (d == -1, return(1));
d = d^2;
);
d == -1
};
MR(n,k)={
for(i=1,k,
if(!sprp(n,random(n-2)+2), return(0))
);
1
};

20
Task/Miller-Rabin-primality-test/PARI-GP/miller-rabin-primality-test-3.parigp
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@ -1,20 +0,0 @@
A006945=[9, 2047, 1373653, 25326001, 3215031751, 2152302898747, 3474749660383, 341550071728321, 341550071728321, 3825123056546413051];
Miller(n)={
if (n%2 == 0, return(n == 2)); \\ Handle even numbers
if (n < 3, return(0)); \\ Handle 0, 1, and negative numbers
if (n < 1<<64,
\\ Feitsma
for(i=1,#A006945,
if (n < A006945[i], return(1));
if(!sprp(n, prime(i)), return(0));
);
sprp(n,31)&sprp(n,37)
,
\\ Miller + Bach
for(b=2,2*log(n)^2,
if(!sprp(n, b), return(0))
);
1
)
};

39
Task/Miller-Rabin-primality-test/PHP/miller-rabin-primality-test.php
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@ -1,39 +0,0 @@
<?php
function is_prime($n, $k) {
if ($n == 2)
return true;
if ($n < 2 || $n % 2 == 0)
return false;
$d = $n - 1;
$s = 0;
while ($d % 2 == 0) {
$d /= 2;
$s++;
}
for ($i = 0; $i < $k; $i++) {
$a = rand(2, $n-1);
$x = bcpowmod($a, $d, $n);
if ($x == 1 || $x == $n-1)
continue;
for ($j = 1; $j < $s; $j++) {
$x = bcmod(bcmul($x, $x), $n);
if ($x == 1)
return false;
if ($x == $n-1)
continue 2;
}
return false;
}
return true;
}
for ($i = 1; $i <= 1000; $i++)
if (is_prime($i, 10))
echo "$i, ";
echo "\n";
?>

57
Task/Miller-Rabin-primality-test/PascalABC.NET/miller-rabin-primality-test.pas
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@ -1,57 +0,0 @@
uses System.Security.Cryptography;
function IsProbablePrime(source: biginteger; certainty: integer): boolean;
begin
if (source = 2) or (source = 3) then
begin result := true; exit end;
if (source < 2) or (source mod 2 = 0) then
begin result := false; exit end;
var d := source - 1;
var s := 0;
while d mod 2 = 0 do
begin
d := d div 2;
s += 1;
end;
var rng := RandomNumberGenerator.Create();
var bytes := new byte[source.ToByteArray.LongLength];
var a: biginteger;
loop certainty do
begin
repeat
rng.GetBytes(bytes);
a := new BigInteger(bytes);
until (a >= 2) and (a < source - 2);
var x := BigInteger.ModPow(a, d, source);
if (x = 1) or (x = source - 1) then continue;
for var r := 1 to s - 1 do
begin
x := BigInteger.ModPow(x, 2, source);
if x = 1 then
begin result := false; exit end;
if x = source - 1 then break;
end;
if x <> source - 1 then
begin result := false; exit end;
end;
result := true;
end;
begin
var data := |'4547337172376300111955330758342147474062293202868155909489',
'4547337172376300111955330758342147474062293202868155909393'|;
foreach var candidate in data do
isprobableprime(candidate.tobiginteger, 10).Println;
foreach var x in (900..1000) do
if isprobableprime(x, 10) then print(x);
end.

32
Task/Miller-Rabin-primality-test/Perl/miller-rabin-primality-test-1.pl
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@ -1,32 +0,0 @@
use bigint try => 'GMP';
sub is_prime {
my ($n, $k) = @_;
return 1 if $n == 2;
return 0 if $n < 2 or $n % 2 == 0;
my $d = $n - 1;
my $s = 0;
while (!($d % 2)) {
$d /= 2;
$s++;
}
LOOP: for (1 .. $k) {
my $a = 2 + int(rand($n - 2));
my $x = $a->bmodpow($d, $n);
next if $x == 1 or $x == $n - 1;
for (1 .. $s - 1) {
$x = ($x * $x) % $n;
return 0 if $x == 1;
next LOOP if $x == $n - 1;
}
return 0;
}
return 1;
}
print join ", ", grep { is_prime $_, 10 } (1 .. 1000);

10
Task/Miller-Rabin-primality-test/Perl/miller-rabin-primality-test-2.pl
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@ -1,10 +0,0 @@
use ntheory qw/is_strong_pseudoprime miller_rabin_random/;
sub is_prime_mr {
my $n = shift;
# If 32-bit, we can do this with 3 bases.
return is_strong_pseudoprime($n, 2, 7, 61) if ($n >> 32) == 0;
# If 64-bit, 7 is all we need.
return is_strong_pseudoprime($n, 2, 325, 9375, 28178, 450775, 9780504, 1795265022) if ($n >> 64) == 0;
# Otherwise, perform a number of random base tests, and the result is a probable prime test.
return miller_rabin_random($n, 20);
}

10
Task/Miller-Rabin-primality-test/Perl/miller-rabin-primality-test-3.pl
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@ -1,10 +0,0 @@
use Math::Primality qw/is_strong_pseudoprime/;
sub is_prime_mr {
my $n = shift;
return 0 if $n < 2;
for (2,3,5,7,11,13,17,19,23,29,31,37) {
return 0 unless $n <= $_ || is_strong_pseudoprime($n,$_);
}
1;
}
for (1..100) { say if is_prime_mr($_) }

69
Task/Miller-Rabin-primality-test/Phix/miller-rabin-primality-test-1.phix
View file

@ -1,69 +0,0 @@
(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">powermod</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- calculate a^n%mod</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">and_bits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">witness</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- n-1 = 2^s * d with d odd by factoring powers of 2 from n-1</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">powermod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">y</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">w</span><span style="color: #0000FF;">=</span><span style="color: #000000;">s</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">w</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">!=</span> <span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">!=</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">y</span>
<span style="color: #000000;">w</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">!=</span> <span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">true</span><span style="color: #0000FF;">;</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">is_prime_mr</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">0</span> <span style="color: #008080;">and</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">0</span> <span style="color: #008080;">and</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">3</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">true</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">;</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">and_bits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">/=</span> <span style="color: #000000;">2</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">1373653</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">9080191</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">31</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">73</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">elsif</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">machine_bits</span><span style="color: #0000FF;">()=</span><span style="color: #000000;">32</span> <span style="color: #008080;">and</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">94910107</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">machine_bits</span><span style="color: #0000FF;">()=</span><span style="color: #000000;">64</span> <span style="color: #008080;">and</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">4295041217</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">61</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">else</span>
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"limits exceeded\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">witness</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">tests</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">999983</span><span style="color: #0000FF;">,</span><span style="color: #000000;">999809</span><span style="color: #0000FF;">,</span><span style="color: #000000;">999727</span><span style="color: #0000FF;">,</span><span style="color: #000000;">52633</span><span style="color: #0000FF;">,</span><span style="color: #000000;">60787</span><span style="color: #0000FF;">,</span><span style="color: #000000;">999999</span><span style="color: #0000FF;">,</span><span style="color: #000000;">999995</span><span style="color: #0000FF;">,</span><span style="color: #000000;">999991</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%d is %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],{</span><span style="color: #008000;">"composite"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"prime"</span><span style="color: #0000FF;">}[</span><span style="color: #000000;">is_prime_mr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--

104
Task/Miller-Rabin-primality-test/Phix/miller-rabin-primality-test-2.phix
View file

@ -1,104 +0,0 @@
(phixonline)-->
<span style="color: #000080;font-style:italic;">-- this is transpiled (then manually copied) to mpz_prime() in mpfr.js:</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">modp47</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">NULL</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">w</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">witness_ranges</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">mpz_prime_mr</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpz</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">10</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- deterministic to 3,317,044,064,679,887,385,961,981</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">primes</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">11</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">13</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">17</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">19</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">23</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">29</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">31</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">37</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">41</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">43</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">47</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">primes</span><span style="color: #0000FF;">[$])<=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_get_integer</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">),</span><span style="color: #000000;">primes</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">modp47</span><span style="color: #0000FF;">=</span><span style="color: #004600;">NULL</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">modp47</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"614_889_782_588_491_410"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- === product(primes), largest &lt; 2^64</span>
<span style="color: #000000;">w</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #000080;font-style:italic;">-- Best known deterministic witnesses for given range and set of bases
-- https://miller-rabin.appspot.com/
-- https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test</span>
<span style="color: #000000;">witness_ranges</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #008000;">"341_531"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"9345883071009581737"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"1_050_535_501"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"336781006125"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"9639812373923155"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"350_269_456_337"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"4230279247111683200"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"14694767155120705706"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"16641139526367750375"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"55_245_642_489_451"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"2"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"141889084524735"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"1199124725622454117"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"11096072698276303650"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"7_999_252_175_582_851"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"2"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"4130806001517"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"149795463772692060"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"186635894390467037"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"3967304179347715805"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"585_226_005_592_931_977"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"2"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"123635709730000"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"9233062284813009"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"43835965440333360"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"761179012939631437"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"1263739024124850375"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"18_446_744_073_709_551_615"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"2"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"325"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"9375"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"28178"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"450775"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"9780504"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"1795265022"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"318_665_857_834_031_151_167_461"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"2"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"3"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"5"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"7"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"11"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"13"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"17"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"19"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"23"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"29"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"31"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"37"</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{</span><span style="color: #008000;">"3_317_044_064_679_887_385_961_981"</span><span style="color: #0000FF;">,{</span><span style="color: #008000;">"2"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"3"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"5"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"7"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"11"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"13"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"17"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"19"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"23"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"29"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"31"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"37"</span><span style="color: #0000FF;">,</span> <span style="color: #008000;">"41"</span><span style="color: #0000FF;">}}}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">2</span><span style="color: #0000FF;">])</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">2</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">2</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">mpz_gcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">w</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">modp47</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">w</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">false</span> <span style="color: #000080;font-style:italic;">-- eliminates 86.2% of all integers</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000080;font-style:italic;">--
-- Choose input witness bases:
--</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">witnesses</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[$][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])>=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">witnesses</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">k</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">mpz_sub_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_rand</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- a := 0..a-1 (cf rand(n) yields 1..n)</span>
<span style="color: #7060A8;">mpz_add_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">witnesses</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">else</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])<</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">witnesses</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">witness_ranges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">exit</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">mpz_sub_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">nm1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- d &gt;&gt;= 4 while (d & 0xf) == 0 # suck out factors of 2
-- (d &gt;&gt;= (d & 3)^2; d &gt;&gt;= (d & 1)^1) if d.even? # 4 bits at a time</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">mpz_even</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_fdiv_q_2exp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">witnesses</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">witnesses</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #7060A8;">mpz_divisible_p</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- skip multiples of input</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">mpz_powm</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- y := b^d % p</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">mpz_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">and</span> <span style="color: #7060A8;">mpz_cmp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">nm1</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">and</span> <span style="color: #7060A8;">mpz_cmp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">nm1</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_powm_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">y</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- y := y^2 mod p</span>
<span style="color: #7060A8;">mpz_mul_2exp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- s &lt;&lt; 1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">nm1</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_even</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #004600;">false</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">true</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<!--

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Task/Miller-Rabin-primality-test/Phix/miller-rabin-primality-test-3.phix
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@ -1,30 +0,0 @@
(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"9223372036854774808"</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">4808</span> <span style="color: #008080;">to</span> <span style="color: #000000;">5808</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">-- (b ends thus)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" %s%s"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">),</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">5</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">]})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">mpz_add_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n%d primes found\n\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">tests</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #008000;">"4547337172376300111955330758342147474062293202868155909489"</span><span style="color: #0000FF;">,</span>
<span style="color: #008000;">"4547337172376300111955330758342147474062293202868155909393"</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_set_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)?</span><span style="color: #008000;">"is prime"</span><span style="color: #0000FF;">:</span><span style="color: #008000;">"is composite"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s %s\n\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">p</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1300</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_ui_pow_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_sub_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"2^%d-1 is prime\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--

47
Task/Miller-Rabin-primality-test/PicoLisp/miller-rabin-primality-test.l
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@ -1,47 +0,0 @@
(de longRand (N)
(use (R D)
(while (=0 (setq R (abs (rand)))))
(until (> R N)
(unless (=0 (setq D (abs (rand))))
(setq R (* R D)) ) )
(% R N) ) )
(de **Mod (X Y N)
(let M 1
(loop
(when (bit? 1 Y)
(setq M (% (* M X) N)) )
(T (=0 (setq Y (>> 1 Y)))
M )
(setq X (% (* X X) N)) ) ) )
(de _prim? (N D S)
(use (A X R)
(while (> 2 (setq A (longRand N))))
(setq R 0 X (**Mod A D N))
(loop
(T
(or
(and (=0 R) (= 1 X))
(= X (dec N)) )
T )
(T
(or
(and (> R 0) (= 1 X))
(>= (inc 'R) S) )
NIL )
(setq X (% (* X X) N)) ) ) )
(de prime? (N K)
(default K 50)
(and
(> N 1)
(bit? 1 N)
(let (D (dec N) S 0)
(until (bit? 1 D)
(setq
D (>> 1 D)
S (inc S) ) )
(do K
(NIL (_prim? N D S))
T ) ) ) )

89
Task/Miller-Rabin-primality-test/Pike/miller-rabin-primality-test.pike
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@ -1,89 +0,0 @@
int pow_mod(int m, int i,int mod){
int x=1,y=m%mod;
while(i){
if(i&1) x = x*y%mod;
y = y*y%mod;
i>>=1;
}
return x;
}
bool mr_pass(int a, int s, int d, int n){
int a_to_power = pow_mod(a, d, n);
if(a_to_power == 1)
return true;
for(int i = 0; i< s - 1; i++){
if(a_to_power == n - 1)
return true;
a_to_power = (a_to_power * a_to_power) % n;
}
return a_to_power == n - 1;
}
int is_prime(int n){
array(int) prime ;
prime = ({2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113});
int idx = search( prime, n);
if( n < 113 && n == prime[idx] ){
return 1;
}
if(n < 2047) prime = ({2, 3});
if(n < 1373653)prime = ({2, 3});
if(n < 9080191)prime = ({31, 73});
if(n < 25326001)prime = ({2, 3, 5});
if(n < 3215031751)prime = ({2, 3, 5, 7});
if(n < 4759123141)prime = ({2, 7, 61});
if(n < 1122004669633)prime = ({2, 13, 23, 1662803});
if(n < 2152302898747)prime = ({2, 3, 5, 7, 11});
if(n < 3474749660383)prime = ({2, 3, 5, 7, 11, 13});
if(n < 341550071728321)prime = ({2, 3, 5, 7, 11, 13, 17});
if(n < 3825123056546413051)prime = ({2, 3, 5, 7, 11, 13, 17, 19, 23});
if(n < 18446744073709551616)prime = ({2, 3, 5, 7, 11, 13, 17, 19, 23, 29});
if(n < 318665857834031151167461)prime = ({2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31});
if(n < 3317044064679887385961981)prime = ({2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37});
else prime = ({2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71});
int d = n - 1;
int s = 0;
while(d % 2 == 0){
d >>= 1;
s += 1;
}
for (int repeat=0 ; repeat < sizeof(prime); repeat++){
int a = prime[repeat];
if(!mr_pass(a, s, d, n)){
return 0;
}
}
return 1;
}
int main() {
array(int) lists;
lists =({35201546659608842026088328007565866231962578784643756647773109869245232364730066609837018108561065242031153677,
10513733234846849736873637829838635104309714688896631127438692162131857778044158273164093838937083421380041997,
24684249032065892333066123534168930441269525239006410135714283699648991959894332868446109170827166448301044689,
76921421106760125285550929240903354966370431827792714920086011488103952094969175731459908117375995349245839343,
32998886283809577512914881459957314326750856532546837122557861905096711274255431870995137546575954361422085081,
30925729459015376480470394633122420870296099995740154747268426836472045107181955644451858184784072167623952123,
14083359469338511572632447718747493405040362318205860500297736061630222431052998057250747900577940212317413063,
10422980533212493227764163121880874101060283221003967986026457372472702684601194916229693974417851408689550781,
36261430139487433507414165833468680972181038593593271409697364115931523786727274410257181186996611100786935727,
15579763548573297857414066649875054392128789371879472432457450095645164702139048181789700140949438093329334293});
for(int i=0;i<sizeof(lists);i++){
int n = lists[i];
int chk = is_prime(n);
if(chk == 1) write(sprintf("%d %s\n",n,"PRIME"));
else write(sprintf("%d %s\n",n,"COMPOSIT"));
}
}
TEST
35201546659608842026088328007565866231962578784643756647773109869245232364730066609837018108561065242031153677 PRIME
10513733234846849736873637829838635104309714688896631127438692162131857778044158273164093838937083421380041997 PRIME
24684249032065892333066123534168930441269525239006410135714283699648991959894332868446109170827166448301044689 PRIME
76921421106760125285550929240903354966370431827792714920086011488103952094969175731459908117375995349245839343 PRIME
32998886283809577512914881459957314326750856532546837122557861905096711274255431870995137546575954361422085081 PRIME
30925729459015376480470394633122420870296099995740154747268426836472045107181955644451858184784072167623952123 PRIME
14083359469338511572632447718747493405040362318205860500297736061630222431052998057250747900577940212317413063 PRIME
10422980533212493227764163121880874101060283221003967986026457372472702684601194916229693974417851408689550781 PRIME
36261430139487433507414165833468680972181038593593271409697364115931523786727274410257181186996611100786935727 PRIME
15579763548573297857414066649875054392128789371879472432457450095645164702139048181789700140949438093329334293 PRIME

83
Task/Miller-Rabin-primality-test/Prolog/miller-rabin-primality-test.pro
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@ -1,83 +0,0 @@
:- module(primality, [is_prime/2]).
% is_prime/2 returns false if N is composite, true if N probably prime
% implements a Miller-Rabin primality test and is deterministic for N < 3.415e+14,
% and is probabilistic for larger N. Adapted from the Erlang version.
is_prime(1, Ret) :- Ret = false, !. % 1 is non-prime
is_prime(2, Ret) :- Ret = true, !. % 2 is prime
is_prime(3, Ret) :- Ret = true, !. % 3 is prime
is_prime(N, Ret) :-
N > 3, (N mod 2 =:= 0), Ret = false, !. % even number > 3 is composite
is_prime(N, Ret) :-
N > 3, (N mod 2 =:= 1), % odd number > 3
N < 341550071728321,
deterministic_witnesses(N, L),
is_mr_prime(N, L, Ret), !. % deterministic test
is_prime(N, Ret) :-
random_witnesses(N, 100, [], Out),
is_mr_prime(N, Out, Ret), !. % probabilistic test
% returns list of deterministic witnesses
deterministic_witnesses(N, L) :- N < 1373653,
L = [2, 3].
deterministic_witnesses(N, L) :- N < 9080191,
L = [31, 73].
deterministic_witnesses(N, L) :- N < 25326001,
L = [2, 3, 5].
deterministic_witnesses(N, L) :- N < 3215031751,
L = [2, 3, 5, 7].
deterministic_witnesses(N, L) :- N < 4759123141,
L = [2, 7, 61].
deterministic_witnesses(N, L) :- N < 1122004669633,
L = [2, 13, 23, 1662803].
deterministic_witnesses(N, L) :- N < 2152302898747,
L = [2, 3, 5, 7, 11].
deterministic_witnesses(N, L) :- N < 3474749660383,
L = [2, 3, 5, 7, 11, 13].
deterministic_witnesses(N, L) :- N < 341550071728321,
L = [2, 3, 5, 7, 11, 13, 17].
% random_witnesses/4 returns a list of K witnesses selected at random with range 2 -> N-2
random_witnesses(_, 0, T, T).
random_witnesses(N, K, T, Out) :-
G is N - 2,
H is 1 + random(G),
I is K - 1,
random_witnesses(N, I, [H | T], Out), !.
% find_ds/2 receives odd integer N and returns [D, S] s.t. N-1 = 2^S * D
find_ds(N, L) :-
A is N - 1,
find_ds(A, 0, L), !.
find_ds(D, S, L) :-
D mod 2 =:= 0,
P is D // 2,
Q is S + 1,
find_ds(P, Q, L), !.
find_ds(D, S, L) :-
L = [D, S].
is_mr_prime(N, As, Ret) :-
find_ds(N, L),
L = [D | T],
T = [S | _],
outer_loop(N, As, D, S, Ret), !.
outer_loop(N, As, D, S, Ret) :-
As = [A | At],
Base is powm(A, D, N),
inner_loop(Base, N, 0, S, Result),
( Result == false -> Ret = false
; Result == true, At == [] -> Ret = true
; outer_loop(N, At, D, S, Ret)
).
inner_loop(Base, N, Loop, S, Result) :-
Next_Base is (Base * Base) mod N,
Next_Loop is Loop + 1,
( Loop =:= 0, Base =:= 1 -> Result = true
; Base =:= N-1 -> Result = true
; Next_Loop =:= S -> Result = false
; inner_loop(Next_Base, N, Next_Loop, S, Result)
).

29
Task/Miller-Rabin-primality-test/PureBasic/miller-rabin-primality-test.basic
View file

@ -1,29 +0,0 @@
Enumeration
#Composite
#Probably_prime
EndEnumeration
Procedure Miller_Rabin(n, k)
Protected d=n-1, s, x, r
If n=2
ProcedureReturn #Probably_prime
ElseIf n%2=0 Or n<2
ProcedureReturn #Composite
EndIf
While d%2=0
d/2
s+1
Wend
While k>0
k-1
x=Int(Pow(2+Random(n-4),d))%n
If x=1 Or x=n-1: Continue: EndIf
For r=1 To s-1
x=(x*x)%n
If x=1: ProcedureReturn #Composite: EndIf
If x=n-1: Break: EndIf
Next
If x<>n-1: ProcedureReturn #Composite: EndIf
Wend
ProcedureReturn #Probably_prime
EndProcedure

39
Task/Miller-Rabin-primality-test/Python/miller-rabin-primality-test-1.py
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@ -1,39 +0,0 @@
import random
def is_Prime(n):
"""
Miller-Rabin primality test.
A return value of False means n is certainly not prime. A return value of
True means n is very likely a prime.
"""
if n!=int(n):
return False
n=int(n)
#Miller-Rabin test for prime
if n==0 or n==1 or n==4 or n==6 or n==8 or n==9:
return False
if n==2 or n==3 or n==5 or n==7:
return True
s = 0
d = n-1
while d%2==0:
d>>=1
s+=1
assert(2**s * d == n-1)
def trial_composite(a):
if pow(a, d, n) == 1:
return False
for i in range(s):
if pow(a, 2**i * d, n) == n-1:
return False
return True
for i in range(8):#number of trials
a = random.randrange(2, n)
if trial_composite(a):
return False
return True

37
Task/Miller-Rabin-primality-test/Python/miller-rabin-primality-test-2.py
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@ -1,37 +0,0 @@
def _try_composite(a, d, n, s):
if pow(a, d, n) == 1:
return False
for i in range(s):
if pow(a, 2**i * d, n) == n-1:
return False
return True # n is definitely composite
def is_prime(n, _precision_for_huge_n=16):
if n in _known_primes:
return True
if any((n % p) == 0 for p in _known_primes) or n in (0, 1):
return False
d, s = n - 1, 0
while not d % 2:
d, s = d >> 1, s + 1
# Returns exact according to http://primes.utm.edu/prove/prove2_3.html
if n < 1373653:
return not any(_try_composite(a, d, n, s) for a in (2, 3))
if n < 25326001:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5))
if n < 118670087467:
if n == 3215031751:
return False
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7))
if n < 2152302898747:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7, 11))
if n < 3474749660383:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7, 11, 13))
if n < 341550071728321:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7, 11, 13, 17))
# otherwise
return not any(_try_composite(a, d, n, s)
for a in _known_primes[:_precision_for_huge_n])
_known_primes = [2, 3]
_known_primes += [x for x in range(5, 1000, 2) if is_prime(x)]

74
Task/Miller-Rabin-primality-test/Quackery/miller-rabin-primality-test.quackery
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@ -1,74 +0,0 @@
[ nextword
dup $ "" = if
[ $ "var needs a name"
message put bail ]
$ " [ stack 0 ] is "
over join
$ " [ "
join over join
$ " replace ] is ->"
join over join
$ " [ "
join over join
$ " share ] is "
join swap join
$ "-> " join
swap join ] builds var ( [ $ --> [ $ )
10000 eratosthenes
[ 1 & not ] is even ( n --> b )
var n var d var s var a
var t var x var y var c
[ dup 10000 < iff isprime done
dup even iff
[ drop false ] done
dup ->n
[ 1 - 0 swap
[ dup even while
1 >> dip 1+ again ] ]
->d ->s
false ->c
20 times
[ n-> 2 - random 2 + ->a
s-> ->t
a-> d-> n-> **mod ->x
s-> times
[ x-> 2 n-> **mod ->y
y-> 1 =
x-> 1 !=
x-> n-> 1 - !=
and and iff
[ true ->c conclude ]
done
y-> ->x ]
c-> iff conclude done
y-> 1 != if
[ true ->c conclude ] ]
c-> not ] is prime ( n --> b )
say "Primes < 100:" cr
100 times [ i^ prime if [ i^ echo sp ] ]
cr cr
[] 1000 times
[ i^ 9223372036854774808 + dup
prime iff join else drop ]
say "There are "
dup size echo
say " primes between 9223372036854774808 and 9223372036854775808:"
cr cr
witheach [ echo i^ 1+ 4 mod iff sp else cr ]
cr cr
4547337172376300111955330758342147474062293202868155909489 dup echo
prime iff
[ say " is prime." ]
else
[ say " is not prime." ]
cr cr
4547337172376300111955330758342147474062293202868155909393 dup echo
prime iff
[ say "is prime." ]
else
[ say " is not prime." ]

31
Task/Miller-Rabin-primality-test/REXX/miller-rabin-primality-test.rexx
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@ -1,31 +0,0 @@
include Settings
say 'MILLER-RABIN PRIMALITY TEST - 4 Mar 2025'
say version
say
numeric digits 1000
say '25 numbers of the form 2^n-1, mostly Mersenne primes'
say 'Up to about 25 digits deterministic, above probabilistic'
say
mm = '2 3 5 7 11 13 17 19 23 31 61 89 97 107 113 127 131 521 ',
|| '607 1279 2203 2281 2293 3217 3221'
do nn = 1 to 25
a = Word(mm,nn); b = 2**a-1; l = Length(b)
call Time('r'); p = Prime(b); e = Time('e')
if l > 20 then
b = Left(b,10)'...'Right(b,10)
select
when p = 0 then
p = 'not'
when l < 26 then
p = 'for sure'
otherwise
p = 'probable'
end
say '2^'a'-1' '=' b '('l' digits) is' p 'prime' '('Format(e,,3) 'seconds)'
end
return
include Functions
include Numbers
include Abend

32
Task/Miller-Rabin-primality-test/Racket/miller-rabin-primality-test.rkt
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@ -1,32 +0,0 @@
#lang racket
(define (miller-rabin-expmod base exp m)
(define (squaremod-with-check x)
(define (check-nontrivial-sqrt1 x square)
(if (and (= square 1)
(not (= x 1))
(not (= x (- m 1))))
0
square))
(check-nontrivial-sqrt1 x (remainder (expt x 2) m)))
(cond ((= exp 0) 1)
((even? exp) (squaremod-with-check
(miller-rabin-expmod base (/ exp 2) m)))
(else
(remainder (* base (miller-rabin-expmod base (- exp 1) m))
m))))
(define (miller-rabin-test n)
(define (try-it a)
(define (check-it x)
(and (not (= x 0)) (= x 1)))
(check-it (miller-rabin-expmod a (- n 1) n)))
(try-it (+ 1 (random (remainder (- n 1) 4294967087)))))
(define (fast-prime? n times)
(for/and ((i (in-range times)))
(miller-rabin-test n)))
(define (prime? n(times 100))
(fast-prime? n times))
(prime? 4547337172376300111955330758342147474062293202868155909489) ;-> outputs true

42
Task/Miller-Rabin-primality-test/Raku/miller-rabin-primality-test.raku
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@ -1,42 +0,0 @@
# the expmod-function from: http://rosettacode.org/wiki/Modular_exponentiation
sub expmod(Int $a is copy, Int $b is copy, $n) {
my $c = 1;
repeat while $b div= 2 {
($c *= $a) %= $n if $b % 2;
($a *= $a) %= $n;
}
$c;
}
subset PrimeCandidate of Int where { $_ > 2 and $_ % 2 };
my Bool multi sub is_prime(Int $n, Int $k) { return False; }
my Bool multi sub is_prime(2, Int $k) { return True; }
my Bool multi sub is_prime(PrimeCandidate $n, Int $k) {
my Int $d = $n - 1;
my Int $s = 0;
while $d %% 2 {
$d div= 2;
$s++;
}
for (2 ..^ $n).pick($k) -> $a {
my $x = expmod($a, $d, $n);
# one could just write "next if $x == 1 | $n - 1"
# but this takes much more time in current rakudo/nom
next if $x == 1 or $x == $n - 1;
for 1 ..^ $s {
$x = $x ** 2 mod $n;
return False if $x == 1;
last if $x == $n - 1;
}
return False if $x !== $n - 1;
}
return True;
}
say (1..1000).grep({ is_prime($_, 10) }).join(", ");

48
Task/Miller-Rabin-primality-test/Ring/miller-rabin-primality-test.ring
View file

@ -1,48 +0,0 @@
# Project : Miller–Rabin primality test
see "Input a number: " give n
see "Input test: " give k
test = millerrabin(n,k)
if test = 0
see "Probably Prime" + nl
else
see "Composite" + nl
ok
func millerrabin(n, k)
if n = 2
millerRabin = 0
return millerRabin
ok
if n % 2 = 0 or n < 2
millerRabin = 1
return millerRabin
ok
d = n - 1
s = 0
while d % 2 = 0
d = d / 2
s = s + 1
end
while k > 0
k = k - 1
base = 2 + floor((random(10)/10)*(n-3))
x = pow(base, d) % n
if x != 1 and x != n-1
for r=1 to s-1
x = (x * x) % n
if x = 1
millerRabin = 1
return millerRabin
ok
if x = n-1
exit
ok
next
if x != n-1
millerRabin = 1
return millerRabin
ok
ok
end

22
Task/Miller-Rabin-primality-test/Ruby/miller-rabin-primality-test-1.rb
View file

@ -1,22 +0,0 @@
def miller_rabin_prime?(n, g)
d = n - 1
s = 0
while d % 2 == 0
d /= 2
s += 1
end
g.times do
a = 2 + rand(n - 4)
x = a.pow(d, n) # x = (a**d) % n
next if x == 1 || x == n - 1
for r in (1..s - 1)
x = x.pow(2, n) # x = (x**2) % n
return false if x == 1
break if x == n - 1
end
return false if x != n - 1
end
true # probably
end
p primes = (3..1000).step(2).find_all {|i| miller_rabin_prime?(i,10)}

7
Task/Miller-Rabin-primality-test/Ruby/miller-rabin-primality-test-2.rb
View file

@ -1,7 +0,0 @@
puts miller_rabin_prime?(94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881,1000)
puts miller_rabin_prime?(138028649176899647846076023812164793645371887571371559091892986639999096471811910222267538577825033963552683101137782650479906670021895135954212738694784814783986671046107023185842481502719762055887490765764329237651328922972514308635045190654896041748716218441926626988737664133219271115413563418353821396401,1000)
puts miller_rabin_prime?(123301261697053560451930527879636974557474268923771832437126939266601921428796348203611050423256894847735769138870460373141723679005090549101566289920247264982095246187318303659027201708559916949810035265951104246512008259674244307851578647894027803356820480862664695522389066327012330793517771435385653616841,1000)
puts miller_rabin_prime?(119432521682023078841121052226157857003721669633106050345198988740042219728400958282159638484144822421840470442893056822510584029066504295892189315912923804894933736660559950053226576719285711831138657839435060908151231090715952576998400120335346005544083959311246562842277496260598128781581003807229557518839,1000)
puts miller_rabin_prime?(132082885240291678440073580124226578272473600569147812319294626601995619845059779715619475871419551319029519794232989255381829366374647864619189704922722431776563860747714706040922215308646535910589305924065089149684429555813953571007126408164577035854428632242206880193165045777949624510896312005014225526731,1000)
puts miller_rabin_prime?(153410708946188157980279532372610756837706984448408515364579602515073276538040155990230789600191915021209039203172105094957316552912585741177975853552299222501069267567888742458519569317286299134843250075228359900070009684517875782331709619287588451883575354340318132216817231993558066067063143257425853927599,1000)
puts miller_rabin_prime?(103130593592068072608023213244858971741946977638988649427937324034014356815504971087381663169829571046157738503075005527471064224791270584831779395959349442093395294980019731027051356344056416276026592333932610954020105156667883269888206386119513058400355612571198438511950152690467372712488391425876725831041,1000)

124
Task/Miller-Rabin-primality-test/Ruby/miller-rabin-primality-test-3.rb
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@ -1,124 +0,0 @@
class Integer
# Returns true if +self+ is a prime number, else returns false.
def primemr?(k = 10)
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
return primes.include? self if self <= primes.last
modp47 = 614_889_782_588_491_410 # => primes.reduce(:*), largest < 2^64
return false if self.gcd(modp47) != 1 # eliminates 86.2% of all integers
# Choose input witness bases: wits = [range, [wit_bases]] or nil
wits = WITNESS_RANGES.find { |range, wits| range > self }
witnesses = wits && wits[1] || k.times.map{ 2 + rand(self - 4) }
miller_rabin_test(witnesses)
end
private
# Returns true if +self+ passes Miller-Rabin Test with witness bases +b+
def miller_rabin_test(witnesses) # use witness list to test with
neg_one_mod = n = d = self - 1 # these are even as 'self' always odd
d >>= 4 while (d & 0xf) == 0 # suck out factors of 2
(d >>= (d & 3)^2; d >>= (d & 1)^1) if d.even? # 4 bits at a time
witnesses.each do |b| # do M-R test with each witness base
next if (b % self) == 0 # **skip base if a multiple of input**
s = d
y = b.pow(d, self) # y = (b**d) mod self
until y == 1 || y == neg_one_mod || s == n
y = y.pow(2, self) # y = (y**2) mod self
s <<= 1
end
return false unless y == neg_one_mod || s.odd?
end
true
end
# Best known deterministic witnesses for given range and set of bases
# https://miller-rabin.appspot.com/
# https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
WITNESS_RANGES = {
341_531 => [9345883071009581737],
1_050_535_501 => [336781006125, 9639812373923155],
350_269_456_337 => [4230279247111683200, 14694767155120705706, 16641139526367750375],
55_245_642_489_451 => [2, 141889084524735, 1199124725622454117, 11096072698276303650],
7_999_252_175_582_851 => [2, 4130806001517, 149795463772692060, 186635894390467037,
3967304179347715805],
585_226_005_592_931_977 => [2, 123635709730000, 9233062284813009, 43835965440333360,
761179012939631437, 1263739024124850375],
18_446_744_073_709_551_615 => [2, 325, 9375, 28178, 450775, 9780504, 1795265022],
318_665_857_834_031_151_167_461 => [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37],
3_317_044_064_679_887_385_961_981 => [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
}
end
def tm; t = Time.now; yield; Time.now - t end
# 10 digit primes
n = 2147483647
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 18 digit non-prime
n = 844674407370955389
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 19 digit primes
n = 9241386435364257883
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 20 digit primes; largest < 2^64
n = 18446744073709551533
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 58 digit prime
n = 4547337172376300111955330758342147474062293202868155909489
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 58 digit non-prime
n = 4547337172376300111955330758342147474062293202868155909393
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 81 digit prime
n = 100000000000000000000000000000000000000000000000000000000000000000000000001309503
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 81 digit non-prime
n = 100000000000000000000000000000000000000000000000000000000000000000000000001309509
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
# 308 digit prime
n = 94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = 138028649176899647846076023812164793645371887571371559091892986639999096471811910222267538577825033963552683101137782650479906670021895135954212738694784814783986671046107023185842481502719762055887490765764329237651328922972514308635045190654896041748716218441926626988737664133219271115413563418353821396401
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = 123301261697053560451930527879636974557474268923771832437126939266601921428796348203611050423256894847735769138870460373141723679005090549101566289920247264982095246187318303659027201708559916949810035265951104246512008259674244307851578647894027803356820480862664695522389066327012330793517771435385653616841
print "\n number = #{n} is prime "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = 119432521682023078841121052226157857003721669633106050345198988740042219728400958282159638484144822421840470442893056822510584029066504295892189315912923804894933736660559950053226576719285711831138657839435060908151231090715952576998400120335346005544083959311246562842277496260598128781581003807229557518839
print "\n number = #{n} is prime "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = 132082885240291678440073580124226578272473600569147812319294626601995619845059779715619475871419551319029519794232989255381829366374647864619189704922722431776563860747714706040922215308646535910589305924065089149684429555813953571007126408164577035854428632242206880193165045777949624510896312005014225526731
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = 153410708946188157980279532372610756837706984448408515364579602515073276538040155990230789600191915021209039203172105094957316552912585741177975853552299222501069267567888742458519569317286299134843250075228359900070009684517875782331709619287588451883575354340318132216817231993558066067063143257425853927599
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = 103130593592068072608023213244858971741946977638988649427937324034014356815504971087381663169829571046157738503075005527471064224791270584831779395959349442093395294980019731027051356344056416276026592333932610954020105156667883269888206386119513058400355612571198438511950152690467372712488391425876725831041
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts
n = 94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881
print "\n number = #{n} is prime? "; print " in ", tm{ print n.primemr? }, " secs"
puts

55
Task/Miller-Rabin-primality-test/Run-BASIC/miller-rabin-primality-test.basic
View file

@ -1,55 +0,0 @@
input "Input a number:";n
input "Input test:";k
test = millerRabin(n,k)
if test = 0 then
print "Probably Prime"
else
print "Composite"
end if
wait
' ----------------------------------------
' Returns
' Composite = 1
' Probably Prime = 0
' ----------------------------------------
FUNCTION millerRabin(n, k)
if n = 2 then
millerRabin = 0 'probablyPrime
goto [funEnd]
end if
if n mod 2 = 0 or n < 2 then
millerRabin = 1 'composite
goto [funEnd]
end if
d = n - 1
while d mod 2 = 0
d = d / 2
s = s + 1
wend
while k > 0
k = k - 1
base = 2 + int(rnd(1)*(n-3))
x = (base^d) mod n
if x <> 1 and x <> n-1 then
for r=1 To s-1
x =(x * x) mod n
if x=1 then
millerRabin = 1 ' composite
goto [funEnd]
end if
if x = n-1 then exit for
next r
if x<>n-1 then
millerRabin = 1 ' composite
goto [funEnd]
end if
end if
wend
[funEnd]
END FUNCTION

163
Task/Miller-Rabin-primality-test/Rust/miller-rabin-primality-test-1.rs
View file

@ -1,163 +0,0 @@
/* Add these lines to the [dependencies] section of your Cargo.toml file:
num = "0.2.0"
rand = "0.6.5"
*/
use num::bigint::BigInt;
use num::bigint::ToBigInt;
// The modular_exponentiation() function takes three identical types
// (which get cast to BigInt), and returns a BigInt:
fn modular_exponentiation<T: ToBigInt>(n: &T, e: &T, m: &T) -> BigInt {
// Convert n, e, and m to BigInt:
let n = n.to_bigint().unwrap();
let e = e.to_bigint().unwrap();
let m = m.to_bigint().unwrap();
// Sanity check: Verify that the exponent is not negative:
assert!(e >= Zero::zero());
use num::traits::{Zero, One};
// As most modular exponentiations do, return 1 if the exponent is 0:
if e == Zero::zero() {
return One::one()
}
// Now do the modular exponentiation algorithm:
let mut result: BigInt = One::one();
let mut base = n % &m;
let mut exp = e;
loop { // Loop until we can return our result.
if &exp % 2 == One::one() {
result *= &base;
result %= &m;
}
if exp == One::one() {
return result
}
exp /= 2;
base *= base.clone();
base %= &m;
}
}
// is_prime() checks the passed-in number against many known small primes.
// If that doesn't determine if the number is prime or not, then the number
// will be passed to the is_rabin_miller_prime() function:
fn is_prime<T: ToBigInt>(n: &T) -> bool {
let n = n.to_bigint().unwrap();
if n.clone() < 2.to_bigint().unwrap() {
return false
}
let small_primes = vec![2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,
103, 107, 109, 113, 127, 131, 137, 139, 149, 151,
157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
211, 223, 227, 229, 233, 239, 241, 251, 257, 263,
269, 271, 277, 281, 283, 293, 307, 311, 313, 317,
331, 337, 347, 349, 353, 359, 367, 373, 379, 383,
389, 397, 401, 409, 419, 421, 431, 433, 439, 443,
449, 457, 461, 463, 467, 479, 487, 491, 499, 503,
509, 521, 523, 541, 547, 557, 563, 569, 571, 577,
587, 593, 599, 601, 607, 613, 617, 619, 631, 641,
643, 647, 653, 659, 661, 673, 677, 683, 691, 701,
709, 719, 727, 733, 739, 743, 751, 757, 761, 769,
773, 787, 797, 809, 811, 821, 823, 827, 829, 839,
853, 857, 859, 863, 877, 881, 883, 887, 907, 911,
919, 929, 937, 941, 947, 953, 967, 971, 977, 983,
991, 997, 1009, 1013];
use num::traits::Zero; // for Zero::zero()
// Check to see if our number is a small prime (which means it's prime),
// or a multiple of a small prime (which means it's not prime):
for sp in small_primes {
let sp = sp.to_bigint().unwrap();
if n.clone() == sp {
return true
} else if n.clone() % sp == Zero::zero() {
return false
}
}
is_rabin_miller_prime(&n, None)
}
// Note: "use bigint::RandBigInt;" (which is needed for gen_bigint_range())
// fails to work in the Rust playground ( https://play.rust-lang.org ).
// Therefore, I'll create my own here:
fn get_random_bigint(low: &BigInt, high: &BigInt) -> BigInt {
if low == high { // base case
return low.clone()
}
let middle = (low.clone() + high) / 2.to_bigint().unwrap();
let go_low: bool = rand::random();
if go_low {
return get_random_bigint(low, &middle)
} else {
return get_random_bigint(&middle, high)
}
}
// k is the number of times for testing (pass in None to use 5 (the default)).
fn is_rabin_miller_prime<T: ToBigInt>(n: &T, k: Option<usize>) -> bool {
let n = n.to_bigint().unwrap();
let k = k.unwrap_or(10); // number of times for testing (defaults to 10)
use num::traits::{Zero, One}; // for Zero::zero() and One::one()
let zero: BigInt = Zero::zero();
let one: BigInt = One::one();
let two: BigInt = 2.to_bigint().unwrap();
// The call to is_prime() should have already checked this,
// but check for two, less than two, and multiples of two:
if n <= one {
return false
} else if n == two {
return true // 2 is prime
} else if n.clone() % &two == Zero::zero() {
return false // even number (that's not 2) is not prime
}
let mut t: BigInt = zero.clone();
let n_minus_one: BigInt = n.clone() - &one;
let mut s = n_minus_one.clone();
while &s % &two == one {
s /= &two;
t += &one;
}
// Try k times to test if our number is non-prime:
'outer: for _ in 0..k {
let a = get_random_bigint(&two, &n_minus_one);
let mut v = modular_exponentiation(&a, &s, &n);
if v == one {
continue 'outer;
}
let mut i: BigInt = zero.clone();
'inner: while &i < &t {
v = (v.clone() * &v) % &n;
if &v == &n_minus_one {
continue 'outer;
}
i += &one;
}
return false;
}
// If we get here, then we have a degree of certainty
// that n really is a prime number, so return true:
true
}

25
Task/Miller-Rabin-primality-test/Rust/miller-rabin-primality-test-2.rs
View file

@ -1,25 +0,0 @@
fn main() {
let n = 1234687;
let result = is_prime(&n);
println!("Q: Is {} prime? A: {}", n, result);
let n = 1234689;
let result = is_prime(&n);
println!("Q: Is {} prime? A: {}", n, result);
let n = BigInt::parse_bytes("123123423463".as_bytes(), 10).unwrap();
let result = is_prime(&n);
println!("Q: Is {} prime? A: {}", n, result);
let n = BigInt::parse_bytes("123123423465".as_bytes(), 10).unwrap();
let result = is_prime(&n);
println!("Q: Is {} prime? A: {}", n, result);
let n = BigInt::parse_bytes("123123423467".as_bytes(), 10).unwrap();
let result = is_prime(&n);
println!("Q: Is {} prime? A: {}", n, result);
let n = BigInt::parse_bytes("123123423469".as_bytes(), 10).unwrap();
let result = is_prime(&n);
println!("Q: Is {} prime? A: {}", n, result);
}

6
Task/Miller-Rabin-primality-test/Scala/miller-rabin-primality-test-1.scala
View file

@ -1,6 +0,0 @@
import scala.math.BigInt
object MillerRabinPrimalityTest extends App {
val (n, certainty )= (BigInt(args(0)), args(1).toInt)
println(s"$n is ${if (n.isProbablePrime(certainty)) "probably prime" else "composite"}")
}

38
Task/Miller-Rabin-primality-test/Scala/miller-rabin-primality-test-2.scala
View file

@ -1,38 +0,0 @@
import scala.annotation.tailrec
import scala.language.{implicitConversions, postfixOps}
import scala.util.Random
object MillerRabin {
implicit def int2Bools(b: Int): Seq[Boolean] = 31 to 0 by -1 map isBitSet(b)
def isBitSet(byte: Int)(bit: Int): Boolean = ((byte >> bit) & 1) == 1
def mod(num: Int, denom: Int) = if (num % denom >= 0) num % denom else (num % denom) + denom
@tailrec
def isSimple(p: Int, s: Int): Boolean = {
if (s == 0) {
true
}
else if (witness(Random.nextInt(p - 1), p)) {
false
}
else {
isSimple(p, s - 1)
}
}
def witness(a: Int, p: Int): Boolean = {
val b: Seq[Boolean] = p - 1
b.foldLeft(1)((d, b) => if (mod(d * d, p) == 1 && d != 1 && d != p - 1) {
return true
} else {
b match {
case true => mod(mod(d*d, p)*a,p)
case false => mod(d*d, p)
}
}) != 1
}
}

43
Task/Miller-Rabin-primality-test/Scheme/miller-rabin-primality-test.scm
View file

@ -1,43 +0,0 @@
#!r6rs
(import (rnrs base (6))
(srfi :27 random-bits))
;; Fast modular exponentiation.
(define (modexpt b e M)
(cond
((zero? e) 1)
((even? e) (modexpt (mod (* b b) M) (div e 2) M))
((odd? e) (mod (* b (modexpt b (- e 1) M)) M))))
;; Return s, d such that d is odd and 2^s * d = n.
(define (split n)
(let recur ((s 0) (d n))
(if (odd? d)
(values s d)
(recur (+ s 1) (div d 2)))))
;; Test whether the number a proves that n is composite.
(define (composite-witness? n a)
(let*-values (((s d) (split (- n 1)))
((x) (modexpt a d n)))
(and (not (= x 1))
(not (= x (- n 1)))
(let try ((r (- s 1)))
(set! x (modexpt x 2 n))
(or (zero? r)
(= x 1)
(and (not (= x (- n 1)))
(try (- r 1))))))))
;; Test whether n > 2 is a Miller-Rabin pseudoprime, k trials.
(define (pseudoprime? n k)
(or (zero? k)
(let ((a (+ 2 (random-integer (- n 2)))))
(and (not (composite-witness? n a))
(pseudoprime? n (- k 1))))))
;; Test whether any integer is prime.
(define (prime? n)
(and (> n 1)
(or (= n 2)
(pseudoprime? n 50))))

46
Task/Miller-Rabin-primality-test/Seed7/miller-rabin-primality-test.seed7
View file

@ -1,46 +0,0 @@
$ include "seed7_05.s7i";
include "bigint.s7i";
const func boolean: millerRabin (in bigInteger: n, in integer: k) is func
result
var boolean: probablyPrime is TRUE;
local
var bigInteger: d is 0_;
var integer: r is 0;
var integer: s is 0;
var bigInteger: a is 0_;
var bigInteger: x is 0_;
var integer: tests is 0;
begin
if n < 2_ or (n > 2_ and not odd(n)) then
probablyPrime := FALSE;
elsif n > 3_ then
d := pred(n);
s := lowestSetBit(d);
d >>:= s;
while tests < k and probablyPrime do
a := rand(2_, pred(n));
x := modPow(a, d, n);
if x <> 1_ and x <> pred(n) then
r := 1;
while r < s and x <> 1_ and x <> pred(n) do
x := modPow(x, 2_, n);
incr(r);
end while;
probablyPrime := x = pred(n);
end if;
incr(tests);
end while;
end if;
end func;
const proc: main is func
local
var bigInteger: number is 0_;
begin
for number range 2_ to 1000_ do
if millerRabin(number, 10) then
writeln(number);
end if;
end for;
end func;

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