Data commit

Task/Jacobi-symbol/00-META.yaml

@@ -0,0 +1,2 @@
+---
+from: http://rosettacode.org/wiki/Jacobi_symbol

Task/Jacobi-symbol/00-TASK.txt

@@ -0,0 +1,13 @@
+The '''[https://en.wikipedia.org/wiki/Jacobi_symbol Jacobi symbol]''' is a multiplicative function that generalizes the Legendre symbol. Specifically, the Jacobi symbol (a | n) equals the product of the Legendre symbols (a | p_i)^(k_i), where n = p_1^(k_1)*p_2^(k_2)*...*p_i^(k_i) and the Legendre symbol (a | p) denotes the value of  a ^ ((p-1)/2) (mod p)
+* (a | p) ≡    1      if a is a square (mod p)
+* (a | p) ≡        -1      if a is not a square (mod p)
+* (a | p) ≡    0      if a ≡ 0 
+
+If n is prime, then the Jacobi symbol (a | n) equals the Legendre symbol (a | n).
+
+;Task:
+Calculate the Jacobi symbol (a | n).
+
+;Reference:
+* [https://en.wikipedia.org/wiki/Jacobi_symbol Wikipedia article on Jacobi symbol].
+

Task/Jacobi-symbol/11l/jacobi-symbol.11l

@@ -0,0 +1,35 @@
+F jacobi(=a, =n)
+   I n <= 0
+      X ValueError(‘'n' must be a positive integer.’)
+   I n % 2 == 0
+      X ValueError(‘'n' must be odd.’)
+   a %= n
+   V result = 1
+   L a != 0
+      L a % 2 == 0
+         a /= 2
+         V n_mod_8 = n % 8
+         I n_mod_8 C (3, 5)
+            result = -result
+      (a, n) = (n, a)
+      I a % 4 == 3 & n % 4 == 3
+         result = -result
+      a %= n
+   I n == 1
+      R result
+   E
+      R 0
+
+print(‘n\k|’, end' ‘’)
+V kmax = 20
+L(k) 0..kmax
+   print(‘#3’.format(k), end' ‘’)
+print("\n----", end' ‘’)
+L(k) 0..kmax
+   print(end' ‘---’)
+print()
+L(n) (1..21).step(2)
+   print(‘#<2 |’.format(n), end' ‘’)
+   L(k) 0..kmax
+      print(‘#3’.format(jacobi(k, n)), end' ‘’)
+   print()

Task/Jacobi-symbol/AWK/jacobi-symbol.awk

@@ -0,0 +1,46 @@
+# syntax: GAWK -f JACOBI_SYMBOL.AWK
+BEGIN {
+    max_n = 29
+    max_a = max_n + 1
+    printf("n\\a")
+    for (i=1; i<=max_a; i++) {
+      printf("%3d",i)
+      underline = underline " --"
+    }
+    printf("\n---%s\n",underline)
+    for (n=1; n<=max_n; n+=2) {
+      printf("%3d",n)
+      for (a=1; a<=max_a; a++) {
+        printf("%3d",jacobi(a,n))
+      }
+      printf("\n")
+    }
+    exit(0)
+}
+function jacobi(a,n,  result,tmp) {
+    if (n%2 == 0) {
+      print("error: 'n' must be a positive odd integer")
+      exit
+    }
+    a %= n
+    result = 1
+    while (a != 0) {
+      while (a%2 == 0) {
+        a /= 2
+        if (n%8 == 3 || n%8 == 5) {
+          result = -result
+        }
+      }
+      tmp = a
+      a = n
+      n = tmp
+      if (a%4 == 3 && n%4 == 3) {
+        result = -result
+      }
+      a %= n
+    }
+    if (n == 1) {
+      return(result)
+    }
+    return(0)
+}

Task/Jacobi-symbol/Action-/jacobi-symbol.action

@@ -0,0 +1,66 @@
+INCLUDE "D2:PRINTF.ACT" ;from the Action! Tool Kit
+
+INT FUNC Jacobi(INT a,n)
+  INT res,tmp
+
+  IF a>=n THEN
+    a=a MOD n
+  FI
+  res=1
+  WHILE a
+  DO
+    WHILE (a&1)=0
+    DO
+      a=a RSH 1
+      tmp=n&7
+      IF tmp=3 OR tmp=5 THEN
+        res=-res
+      FI
+    OD
+    tmp=a a=n n=tmp
+    IF (a%3)=3 AND (n%3)=3 THEN
+      res=-res
+    FI
+    a=a MOD n
+  OD
+
+  IF n=1 THEN
+    RETURN (res)
+  FI
+RETURN (0)
+
+PROC PrintTable(INT maxK,maxN)
+  INT res,n,k
+  CHAR ARRAY t(10)
+
+  Put('n) Put(7) Put('k) Put(124)
+  FOR k=0 TO maxK
+  DO
+    StrI(k,t) PrintF("%3S",t)
+  OD
+  PutE()
+
+  Put(18) Put(18) Put(18) Put(19)
+  FOR k=0 TO 3*maxK+2
+  DO
+    Put(18)
+  OD
+  PutE()
+
+  FOR n=1 TO maxN STEP 2
+  DO
+    StrI(n,t) PrintF("%3S",t)
+    Put(124)
+    FOR k=0 TO maxK
+    DO
+      res=Jacobi(k,n)
+      StrI(res,t) PrintF("%3S",t)
+    OD
+    PutE()
+  OD
+RETURN
+
+PROC Main()
+  Put(125) PutE() ;clear the screen
+  PrintTable(10,39)
+RETURN

Task/Jacobi-symbol/Arturo/jacobi-symbol.arturo

@@ -0,0 +1,30 @@
+jacobi: function [n,k][
+    N: n % k
+    K: k
+
+    result: 1
+    while [N <> 0][
+        while [even? N][
+            N: shr N 1
+            if contains? [3 5] and K 7 ->
+                result: neg result
+        ]
+        [N,K]: @[K,N]
+        if and? 3=and N 3 3=and K 3 ->
+            result: neg result
+        N: N % K
+    ]
+    if K <> 1 ->
+        result: 0
+
+    return result
+]
+
+print ["" "k/n" "|"] ++ map to [:string] 1..20 'item -> pad.left item 2
+print repeat "=" 67
+loop range.step:2 1 21 'k [
+    print [
+        "" pad to :string k 3 "|" join.with:" " map to [:string] map 1..20 'n -> jacobi n k
+                                                                        'item -> pad.left item 2
+    ]
+]

Task/Jacobi-symbol/AutoHotkey/jacobi-symbol.ahk

@@ -0,0 +1,33 @@
+result := "n/k|"
+loop 20
+    result .= SubStr(" " A_Index, -1) " "
+l := StrLen(result)
+result .= "`n"
+loop % l
+    result .= "-"
+result .= "`n"
+
+loop 21
+{
+    if !Mod(n := A_Index, 2)
+        continue
+    result .= SubStr(" " n, -1) " |"
+    loop 20
+        result .= SubStr(" " jacobi(a := A_Index, n), -1) " "
+    result .= "`n"
+}
+MsgBox, 262144, , % result
+return
+
+jacobi(a, n) {
+    a := Mod(a, n), t := 1
+    while (a != 0) {
+        while !Mod(a, 2)
+            a := a >> 1, r := Mod(n, 8), t := (r=3 || r=5) ? -t : t
+        r := n, n := a, a := r
+        if (Mod(a, 4)=3 && Mod(n, 4)=3)
+            t := -t
+        a := Mod(a, n)
+    }
+    return (n=1) ? t : 0
+}

Task/Jacobi-symbol/C++/jacobi-symbol.cpp

@@ -0,0 +1,44 @@
+#include <algorithm>
+#include <cassert>
+#include <iomanip>
+#include <iostream>
+
+int jacobi(int n, int k) {
+    assert(k > 0 && k % 2 == 1);
+    n %= k;
+    int t = 1;
+    while (n != 0) {
+        while (n % 2 == 0) {
+            n /= 2;
+            int r = k % 8;
+            if (r == 3 || r == 5)
+                t = -t;
+        }
+        std::swap(n, k);
+        if (n % 4 == 3 && k % 4 == 3)
+            t = -t;
+        n %= k;
+    }
+    return k == 1 ? t : 0;
+}
+
+void print_table(std::ostream& out, int kmax, int nmax) {
+    out << "n\\k|";
+    for (int k = 0; k <= kmax; ++k)
+        out << ' ' << std::setw(2) << k;
+    out << "\n----";
+    for (int k = 0; k <= kmax; ++k)
+        out << "---";
+    out << '\n';
+    for (int n = 1; n <= nmax; n += 2) {
+        out << std::setw(2) << n << " |";
+        for (int k = 0; k <= kmax; ++k)
+            out << ' ' << std::setw(2) << jacobi(k, n);
+        out << '\n';
+    }
+}
+
+int main() {
+    print_table(std::cout, 20, 21);
+    return 0;
+}

Task/Jacobi-symbol/C/jacobi-symbol.c

@@ -0,0 +1,39 @@
+#include <stdlib.h>
+#include <stdio.h>
+
+#define SWAP(a, b) (((a) ^= (b)), ((b) ^= (a)), ((a) ^= (b)))
+
+int jacobi(unsigned long a, unsigned long n) {
+	if (a >= n) a %= n;
+	int result = 1;
+	while (a) {
+		while ((a & 1) == 0) {
+			a >>= 1;
+			if ((n & 7) == 3 || (n & 7) == 5) result = -result;
+		}
+		SWAP(a, n);
+		if ((a & 3) == 3 && (n & 3) == 3) result = -result;
+		a %= n;
+	}
+	if (n == 1) return result;
+	return 0;
+}
+
+void print_table(unsigned kmax, unsigned nmax) {
+	printf("n\\k|");
+	for (int k = 0; k <= kmax; ++k) printf("%'3u", k);
+	printf("\n----");
+	for (int k = 0; k <= kmax; ++k) printf("---");
+	putchar('\n');
+	for (int n = 1; n <= nmax; n += 2) {
+		printf("%-2u |", n);
+		for (int k = 0; k <= kmax; ++k)
+			printf("%'3d", jacobi(k, n));
+		putchar('\n');
+	}
+}
+
+int main() {
+	print_table(20, 21);
+	return 0;
+}

Task/Jacobi-symbol/Crystal/jacobi-symbol.crystal

@@ -0,0 +1,22 @@
+def jacobi(a, n)
+  raise ArgumentError.new "n must b positive and odd" if n < 1 || n.even?
+  res = 1
+  until (a %= n) == 0
+    while a.even?
+      a >>= 1
+      res = -res if [3, 5].includes? n % 8
+    end
+    a, n = n, a
+    res = -res if a % 4 == n % 4 == 3
+  end
+  n == 1 ? res : 0
+end
+
+puts "Jacobian symbols for jacobi(a, n)"
+puts "n\\a  0  1  2  3  4  5  6  7  8  9 10"
+puts "------------------------------------"
+1.step(to: 17, by: 2) do |n|
+   printf("%2d ", n)
+   (0..10).each { |a| printf(" % 2d", jacobi(a, n)) }
+   puts
+end

Task/Jacobi-symbol/Erlang/jacobi-symbol.erl

@@ -0,0 +1,27 @@
+jacobi(_, N) when N =< 0 -> jacobi_domain_error;
+jacobi(_, N) when (N band 1) =:= 0 -> jacobi_domain_error;
+jacobi(A, N) when A < 0 ->
+    J2 = ja(-A, N),
+    case N band 3 of
+        1 -> J2;
+        3 -> -J2
+    end;
+jacobi(A, N) -> ja(A, N).
+
+ja(0, _) -> 0;
+ja(1, _) -> 1;
+ja(A, N) when A >= N -> ja(A rem N, N);
+ja(A, N) when (A band 1) =:= 0 -> % A is even
+    J2 = ja(A bsr 1, N),
+    case N band 7 of
+        1 -> J2;
+        3 -> -J2;
+        5 -> -J2;
+        7 -> J2
+    end;
+ja(A, N) ->    % if we get here, A is odd, so we can flip it.
+    J2 = ja(N, A),
+    case (A band 3 =:= 3) and (N band 3 =:= 3) of
+        true  -> -J2;
+        false -> J2
+    end.

Task/Jacobi-symbol/F-Sharp/jacobi-symbol.fs

@@ -0,0 +1,8 @@
+//Jacobi Symbol. Nigel Galloway: July 14th., 2020
+let J n m=let rec J n m g=match n with
+                           0->if m=1 then g else 0
+                          |n when n%2=0->J(n/2) m (if m%8=3 || m%8=5 then -g else g)
+                          |n->J (m%n) n (if m%4=3 && n%4=3 then -g else g)
+          J (n%m) m 1
+printfn "n\m   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30\n     ----------------------------------------------------------------------------------------------------------------------"
+[1..2..29]|>List.iter(fun m->printf "%3d" m; [1..30]|>List.iter(fun n->printf "%4d" (J n m)); printfn "")

Task/Jacobi-symbol/FreeBASIC/jacobi-symbol.basic

@@ -0,0 +1,45 @@
+function gcdp( a as uinteger, b as uinteger ) as uinteger
+    if b = 0 then return a
+    return gcdp( b, a mod b )
+end function
+
+function gcd(a as integer, b as integer) as uinteger
+    return gcdp( abs(a), abs(b) )
+end function
+
+function jacobi( a as uinteger, n as uinteger ) as integer
+    if gcd(a, n)<>1 then return 0
+    if a = 1 then return 1
+    if a>n then return jacobi( a mod n, n )
+    if a mod 2 = 0 then
+        if n mod 8 = 1 or n mod 8 = 7 then
+            return jacobi(a/2, n)
+        else
+            return -jacobi(a/2, n)
+        end if
+    end if
+    dim as integer q = (-1)^((a-1)/2 * (n-1)/2)
+    return q/jacobi(n, a)
+end function
+
+'print a table
+
+function padto( i as ubyte, j as integer ) as string
+    return wspace(i-len(str(j)))+str(j)
+end function
+
+dim as uinteger pn, k, prime(0 to 16) = {3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61}
+dim as string outstr = "  k       "
+
+for k = 1 to 36
+    outstr = outstr + padto(2, k)+"  "
+next k
+print outstr
+print " n"
+for pn=0 to 16
+    outstr= " "+padto( 2, prime(pn) )+"       "
+    for k = 1 to 36
+        outstr = outstr + padto(2, jacobi(k, prime(pn))) + "  "
+    next k
+    print outstr
+next pn

Task/Jacobi-symbol/Go/jacobi-symbol.go

@@ -0,0 +1,60 @@
+package main
+
+import (
+    "fmt"
+    "log"
+    "math/big"
+)
+
+func jacobi(a, n uint64) int {
+    if n%2 == 0 {
+        log.Fatal("'n' must be a positive odd integer")
+    }
+    a %= n
+    result := 1
+    for a != 0 {
+        for a%2 == 0 {
+            a /= 2
+            nn := n % 8
+            if nn == 3 || nn == 5 {
+                result = -result
+            }
+        }
+        a, n = n, a
+        if a%4 == 3 && n%4 == 3 {
+            result = -result
+        }
+        a %= n
+    }
+    if n == 1 {
+        return result
+    }
+    return 0
+}
+
+func main() {
+    fmt.Println("Using hand-coded version:")
+    fmt.Println("n/a  0  1  2  3  4  5  6  7  8  9")
+    fmt.Println("---------------------------------")
+    for n := uint64(1); n <= 17; n += 2 {
+        fmt.Printf("%2d ", n)
+        for a := uint64(0); a <= 9; a++ {
+            fmt.Printf(" % d", jacobi(a, n))
+        }
+        fmt.Println()
+    }
+
+    ba, bn := new(big.Int), new(big.Int)
+    fmt.Println("\nUsing standard library function:")
+    fmt.Println("n/a  0  1  2  3  4  5  6  7  8  9")
+    fmt.Println("---------------------------------")
+    for n := uint64(1); n <= 17; n += 2 {
+        fmt.Printf("%2d ", n)
+        for a := uint64(0); a <= 9; a++ {
+            ba.SetUint64(a)
+            bn.SetUint64(n)
+            fmt.Printf(" % d", big.Jacobi(ba, bn))
+        }
+        fmt.Println()
+    }
+}

Task/Jacobi-symbol/Haskell/jacobi-symbol-1.hs

@@ -0,0 +1,14 @@
+jacobi :: Integer -> Integer -> Integer
+jacobi 0 1 = 1
+jacobi 0 _ = 0
+jacobi a n =
+  let a_mod_n = rem a n
+  in if even a_mod_n
+       then case rem n 8 of
+              1 -> jacobi (div a_mod_n 2) n
+              3 -> negate $ jacobi (div a_mod_n 2) n
+              5 -> negate $ jacobi (div a_mod_n 2) n
+              7 -> jacobi (div a_mod_n 2) n
+       else if rem a_mod_n 4 == 3 && rem n 4 == 3
+              then negate $ jacobi n a_mod_n
+              else jacobi n a_mod_n

Task/Jacobi-symbol/Haskell/jacobi-symbol-2.hs

@@ -0,0 +1,72 @@
+import Data.Bool (bool)
+import Data.List (replicate, transpose)
+import Data.List.Split (chunksOf)
+
+---------------------- JACOBI SYMBOL ---------------------
+
+jacobi :: Int -> Int -> Int
+jacobi = go
+  where
+    go 0 1 = 1
+    go 0 _ = 0
+    go x y
+      | even r =
+          plusMinus
+            (rem y 8 `elem` [3, 5])
+            (go (div r 2) y)
+      | otherwise = plusMinus (p r && p y) (go y r)
+      where
+        plusMinus = bool id negate
+        p = (3 ==) . flip rem 4
+        r = rem x y
+
+
+--------------------------- TEST -------------------------
+main :: IO ()
+main = putStrLn $ jacobiTable 11 9
+
+------------------------- DISPLAY ------------------------
+jacobiTable :: Int -> Int -> String
+jacobiTable nCols nRows =
+  let rowLabels = [1, 3 .. (2 * nRows)]
+      colLabels = [0 .. pred nCols]
+   in withColumnLabels ("" : fmap show colLabels) $
+        labelledRows (fmap show rowLabels) $
+          paddedCols $
+            chunksOf nRows $
+              uncurry jacobi
+                <$> ((,) <$> colLabels <*> rowLabels)
+
+------------------- TABULATION FUNCTIONS -----------------
+paddedCols ::
+  Show a =>
+  [[a]] ->
+  [[String]]
+paddedCols cols =
+  let scols = fmap show <$> cols
+      w = maximum $ length <$> concat scols
+   in map (justifyRight w ' ') <$> scols
+
+labelledRows :: [String] -> [[String]] -> [[String]]
+labelledRows labels cols =
+  let w = maximum $ map length labels
+   in zipWith
+        (:)
+        ((<> " ->") . justifyRight w ' ' <$> labels)
+        (transpose cols)
+
+withColumnLabels :: [String] -> [[String]] -> String
+withColumnLabels _ [] = ""
+withColumnLabels labels rows@(x : _) =
+  let labelRow =
+        unwords $
+          zipWith
+            (`justifyRight` ' ')
+            (length <$> x)
+            labels
+   in unlines $
+        labelRow :
+        replicate (length labelRow) '-' : fmap unwords rows
+
+justifyRight :: Int -> a -> [a] -> [a]
+justifyRight n c = (drop . length) <*> (replicate n c <>)

Task/Jacobi-symbol/J/jacobi-symbol.j

@@ -0,0 +1,14 @@
+NB. functionally equivalent translation of the Lua program found
+NB. at https://en.wikipedia.org/wiki/Jacobi_symbol
+jacobi=: {{
+  assert. (0<x) * 1=2|x
+  y=. x|y
+  t=. 1
+  while. y do.
+    e=. (|.#:y) i.1
+    y=. <.y%2^e
+    t=. t*_1^(*/3 = 4|x,y)+(2|e)*(8|x) e.3 5
+    'x y'=. y, y|x
+  end.
+  t*x=1
+}}"0

Task/Jacobi-symbol/Java/jacobi-symbol.java

@@ -0,0 +1,49 @@
+public class JacobiSymbol {
+
+    public static void main(String[] args) {
+        int max = 30;
+        System.out.printf("n\\k ");
+        for ( int k = 1 ; k <= max ; k++ ) {
+            System.out.printf("%2d  ", k);
+        }
+        System.out.printf("%n");
+        for ( int n = 1 ; n <= max ; n += 2 ) {
+            System.out.printf("%2d  ", n);
+            for ( int k = 1 ; k <= max ; k++ ) {
+                System.out.printf("%2d  ", jacobiSymbol(k, n));
+            }
+            System.out.printf("%n");
+        }
+    }
+
+
+    //  Compute (k n), where k is numerator
+    private static int jacobiSymbol(int k, int n) {
+        if ( k < 0 || n % 2 == 0 ) {
+            throw new IllegalArgumentException("Invalid value. k = " + k + ", n = " + n);
+        }
+        k %= n;
+        int jacobi = 1;
+        while ( k > 0 ) {
+            while ( k % 2 == 0 ) {
+                k /= 2;
+                int r = n % 8;
+                if ( r == 3 || r == 5 ) {
+                    jacobi = -jacobi;
+                }
+            }
+            int temp = n;
+            n = k;
+            k = temp;
+            if ( k % 4 == 3 && n % 4 == 3 ) {
+                jacobi = -jacobi;
+            }
+            k %= n;
+        }
+        if ( n == 1 ) {
+            return jacobi;
+        }
+        return 0;
+    }
+
+}

Task/Jacobi-symbol/Jq/jacobi-symbol.jq

@@ -0,0 +1,21 @@
+def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
+def rpad($len): tostring | ($len - length) as $l | . + (" " * $l)[:$l];
+
+def jacobi(a; n):
+  {a: (a % n), n: n, result: 1}
+  | until(.a == 0;
+          until( .a % 2 != 0;
+                 .a /= 2
+                 | if (.n % 8) | IN(3, 5) then .result *= -1 else . end )
+          | {a: .n, n: .a, result}   # swap .a and .n
+	  | (.n % 4) as $nmod4
+          | if (.a % 4) == $nmod4 and $nmod4 == 3 then .result *= -1 else . end
+          | .a = .a % .n )
+  | if .n == 1 then .result else 0 end ;
+
+"                Table of jacobi(a; n)",
+"n\\k   1   2   3   4   5   6   7   8   9  10  11  12",
+"_____________________________________________________",
+(range( 1; 32; 2) as $n
+ | "\($n|rpad(3))" + reduce range(1; 13) as $a (""; . + (jacobi($a; $n) | lpad(4) ))
+ )

Task/Jacobi-symbol/Julia/jacobi-symbol.julia

@@ -0,0 +1,23 @@
+function jacobi(a, n)
+    a %= n
+    result = 1
+    while a != 0
+        while iseven(a)
+            a ÷= 2
+            ((n % 8) in [3, 5]) && (result *= -1)
+        end
+        a, n = n, a
+        (a % 4 == n % 4 == 3) && (result *= -1)
+        a %= n
+    end
+    return n == 1 ? result : 0
+end
+
+print(" Table of jacobi(a, n) for a 1 to 12, n 1 to 31\n   1   2   3   4   5   6   7   8",
+    "   9  10  11  12\nn\n_____________________________________________________")
+for n in 1:2:31
+    print("\n", rpad(n, 3))
+    for a in 1:11
+        print(lpad(jacobi(a, n), 4))
+    end
+end

Task/Jacobi-symbol/Kotlin/jacobi-symbol.kotlin

@@ -0,0 +1,23 @@
+fun jacobi(A: Int, N: Int): Int {
+    assert(N > 0 && N and 1 == 1)
+    var a = A % N
+    var n = N
+    var result = 1
+    while (a != 0) {
+        var aMod4 = a and 3
+        while (aMod4 == 0) {    // remove factors of four
+            a = a shr 2
+            aMod4 = a and 3
+        }
+        if (aMod4 == 2) {       // if even
+            a = a shr 1         // remove factor 2 and possibly change sign
+            if ((n and 7).let { it == 3 || it == 5 })
+                result = -result
+            aMod4 = a and 3
+        }
+        if (aMod4 == 3 && n and 3 == 3)
+            result = -result
+        a = (n % a).also { n = a }
+    }
+    return if (n == 1) result else 0
+}

Task/Jacobi-symbol/Mathematica/jacobi-symbol.math

@@ -0,0 +1,3 @@
+TableForm[Table[JacobiSymbol[n, k], {n, 1, 17, 2}, {k, 16}],
+ TableHeadings -> {ReplacePart[Range[1, 17, 2], 1 -> "n=1"],
+   ReplacePart[Range[16], 1 -> "k=1"]}]

Task/Jacobi-symbol/Nim/jacobi-symbol.nim

@@ -0,0 +1,34 @@
+template isOdd(n: int): bool = (n and 1) != 0
+template isEven(n: int): bool = (n and 1) == 0
+
+
+func jacobi(n, k: int): range[-1..1] =
+  assert k > 0 and k.isOdd
+  var n = n mod k
+  var k = k
+  result = 1
+  while n != 0:
+    while n.isEven:
+      n = n shr 1
+      if (k and 7) in [3, 5]:
+        result = -result
+    swap n, k
+    if (n and 3) == 3 and (k and 3) == 3:
+      result = -result
+    n = n mod k
+  if k != 1: result = 0
+
+when isMainModule:
+
+  import strutils
+
+  stdout.write "n/k|"
+  for n in 1..20:
+    stdout.write align($n, 3)
+  echo '\n' & repeat("—", 64)
+
+  for k in countup(1, 21, 2):
+    stdout.write align($k, 2), " |"
+    for n in 1..20:
+      stdout.write align($jacobi(n, k), 3)
+    echo ""

Task/Jacobi-symbol/Perl/jacobi-symbol.pl

@@ -0,0 +1,33 @@
+use strict;
+use warnings;
+
+sub J {
+    my($k,$n) = @_;
+
+    $k %= $n;
+    my $jacobi = 1;
+    while ($k) {
+        while (0 == $k % 2) {
+            $k = int $k / 2;
+            $jacobi *= -1 if $n%8 == 3 or $n%8 == 5;
+        }
+        ($k, $n) = ($n, $k);
+        $jacobi *= -1 if $n%4 == 3 and $k%4 == 3;
+        $k %= $n;
+    }
+    $n == 1 ? $jacobi : 0
+}
+
+my $maxa = 1 + (my $maxn = 29);
+
+print 'n\k';
+printf '%4d', $_ for 1..$maxa;
+print "\n";
+print '   ' . '-' x (4 * $maxa) . "\n";
+
+for my $n (1..$maxn) {
+    next if 0 == $n % 2;
+    printf '%3d', $n;
+    printf '%4d', J($_, $n) for 1..$maxa;
+    print "\n"
+}

Task/Jacobi-symbol/Phix/jacobi-symbol.phix

@@ -0,0 +1,27 @@
+(phixonline)-->
+ <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
+ <span style="color: #008080;">function</span> <span style="color: #000000;">jacobi</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
+     <span style="color: #004080;">atom</span> <span style="color: #000000;">result</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
+     <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
+     <span style="color: #008080;">while</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span>
+         <span style="color: #008080;">while</span> <span style="color: #7060A8;">remainder</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;">0</span> <span style="color: #008080;">do</span>
+             <span style="color: #000000;">a</span> <span style="color: #0000FF;">/=</span> <span style="color: #000000;">2</span>
+             <span style="color: #008080;">if</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">8</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: #008080;">then</span> <span style="color: #000000;">result</span> <span style="color: #0000FF;">*=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
+         <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
+         <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">}</span>
+         <span style="color: #008080;">if</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">3</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">3</span> <span style="color: #008080;">then</span> <span style="color: #000000;">result</span> <span style="color: #0000FF;">*=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
+         <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</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: #008080;">iff</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: #000000;">result</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;">function</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\\a   0   1   2   3   4   5   6   7   8   9  10  11\n"</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;">"   ________________________________________________\n"</span><span style="color: #0000FF;">)</span>
+ <span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">31</span> <span style="color: #008080;">by</span> <span style="color: #000000;">2</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;">"%3d"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
+     <span style="color: #008080;">for</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">11</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;">"%4d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">jacobi</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</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"</span><span style="color: #0000FF;">)</span>
+ <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
+<!--

Task/Jacobi-symbol/Python/jacobi-symbol.py

@@ -0,0 +1,21 @@
+def jacobi(a, n):
+    if n <= 0:
+        raise ValueError("'n' must be a positive integer.")
+    if n % 2 == 0:
+        raise ValueError("'n' must be odd.")
+    a %= n
+    result = 1
+    while a != 0:
+        while a % 2 == 0:
+            a /= 2
+            n_mod_8 = n % 8
+            if n_mod_8 in (3, 5):
+                result = -result
+        a, n = n, a
+        if a % 4 == 3 and n % 4 == 3:
+            result = -result
+        a %= n
+    if n == 1:
+        return result
+    else:
+        return 0

Task/Jacobi-symbol/REXX/jacobi-symbol.rexx

@@ -0,0 +1,34 @@
+/*REXX pgm computes/displays the Jacobi symbol, the # of rows & columns can be specified*/
+parse arg rows cols .                            /*obtain optional arguments from the CL*/
+if rows='' | rows==","  then rows= 17            /*Not specified?  Then use the default.*/
+if cols='' | cols==","  then cols= 16            /* "      "         "   "   "      "   */
+call hdrs                                        /*display the (two) headers to the term*/
+      do r=1  by 2  to rows;     _= right(r, 3)  /*build odd (numbered) rows of a table.*/
+                         do c=0  to cols         /* [↓]  build a column for a table row.*/
+                         _= _ ! right(jacobi(c, r), 2);   != '│'  /*reset grid end char.*/
+                         end   /*c*/
+      say _ '║';  != '║'                         /*display a table row; reset grid glyph*/
+      end   /*r*/
+say translate(@.2, '╩╧╝', "╬╤╗")                 /*display the bottom of the grid border*/
+exit                                             /*stick a fork in it,  we're all done. */
+/*──────────────────────────────────────────────────────────────────────────────────────*/
+hdrs: @.1= 'n/a ║';      do c=0  to cols;    @.1= @.1 || right(c, 3)"  ";   end
+      L= length(@.1);                        @.1= left(@.1,   L - 1)    ;          say @.1
+      @.2= '════╬';      do c=0  to cols;    @.2= @.2 || "════╤"        ;   end
+      L= length(@.2);                        @.2= left(@.2,   L - 1)"╗" ;          say @.2
+      != '║'        ;    return                  /*define an external grid border glyph.*/
+/*──────────────────────────────────────────────────────────────────────────────────────*/
+jacobi: procedure; parse arg a,n;  er= '***error***';        $ = 1      /*define result.*/
+        if n//2==0  then do;   say er    n   " must be a positive odd integer.";   exit 13
+                         end
+        a= a // n                                      /*obtain  A  modulus  N          */
+          do while a\==0                               /*perform while  A  isn't zero.  */
+                         do while a//2==0;  a= a % 2   /*divide  A  (as a integer) by 2 */
+                         if n//8==3 | n//8==5  then $= -$               /*use  N mod  8 */
+                         end   /*while a//2==0*/
+          parse value  a  n     with     n  a          /*swap values of variables:  A N */
+          if a//4==3 & n//4==3  then $= -$             /* A mod 4, N mod 4.   Both ≡ 3 ?*/
+          a= a // n                                    /*obtain  A  modulus  N          */
+          end   /*while a\==0*/
+        if n==1  then return $
+                      return 0

Task/Jacobi-symbol/Raku/jacobi-symbol.raku

@@ -0,0 +1,30 @@
+# Jacobi function
+sub infix:<J> (Int $k is copy, Int $n is copy where * % 2) {
+    $k %= $n;
+    my $jacobi = 1;
+    while $k {
+        while $k %% 2 {
+            $k div= 2;
+            $jacobi *= -1 if $n % 8 == 3 | 5;
+        }
+        ($k, $n) = $n, $k;
+        $jacobi *= -1 if 3 == $n%4 & $k%4;
+        $k %= $n;
+    }
+    $n == 1 ?? $jacobi !! 0
+}
+
+# Testing
+
+my $maxa = 30;
+my $maxn = 29;
+
+say 'n\k ', (1..$maxa).fmt: '%3d';
+say '   ', '-' x 4 * $maxa;
+for 1,*+2 … $maxn -> $n {
+    print $n.fmt: '%3d';
+    for 1..$maxa -> $k {
+        print ($k J $n).fmt: '%4d';
+    }
+    print "\n";
+}

Task/Jacobi-symbol/Ruby/jacobi-symbol.rb

@@ -0,0 +1,22 @@
+def jacobi(a, n)
+  raise ArgumentError.new "n must b positive and odd" if n < 1 || n.even?
+  res = 1
+  until (a %= n) == 0
+    while a.even?
+      a >>= 1
+      res = -res if [3, 5].include? n % 8
+    end
+    a, n = n, a
+    res = -res if [a % 4, n % 4] == [3, 3]
+  end
+  n == 1 ? res : 0
+end
+
+puts "Jacobian symbols for jacobi(a, n)"
+puts "n\\a  0  1  2  3  4  5  6  7  8  9 10"
+puts "------------------------------------"
+1.step(to: 17, by: 2) do |n|
+   printf("%2d ", n)
+   (0..10).each { |a| printf(" % 2d", jacobi(a, n)) }
+   puts
+end

Task/Jacobi-symbol/Rust/jacobi-symbol.rust

@@ -0,0 +1,47 @@
+fn jacobi(mut n: i32, mut k: i32) -> i32 {
+    assert!(k > 0 && k % 2 == 1);
+    n %= k;
+    let mut t = 1;
+    while n != 0 {
+        while n % 2 == 0 {
+            n /= 2;
+            let r = k % 8;
+            if r == 3 || r == 5 {
+                t = -t;
+            }
+        }
+        std::mem::swap(&mut n, &mut k);
+        if n % 4 == 3 && k % 4 == 3 {
+            t = -t;
+        }
+        n %= k;
+    }
+    if k == 1 {
+        t
+    } else {
+        0
+    }
+}
+
+fn print_table(kmax: i32, nmax: i32) {
+    print!("n\\k|");
+    for k in 0..=kmax {
+        print!(" {:2}", k);
+    }
+    print!("\n----");
+    for _ in 0..=kmax {
+        print!("---");
+    }
+    println!();
+    for n in (1..=nmax).step_by(2) {
+        print!("{:2} |", n);
+        for k in 0..=kmax {
+            print!(" {:2}", jacobi(k, n));
+        }
+        println!();
+    }
+}
+
+fn main() {
+    print_table(20, 21);
+}

Task/Jacobi-symbol/Scala/jacobi-symbol.scala

@@ -0,0 +1,32 @@
+def jacobi(a_p: Int, n_p: Int): Int =
+{
+    var a = a_p
+    var n = n_p
+    if (n <= 0) return -1
+    if (n % 2 == 0) return -1
+
+    a %= n
+    var result = 1
+    while (a != 0) {
+      while (a % 2 == 0) {
+        a /= 2
+        if (n % 8 == 3 || n % 8 == 5) result = -result
+      }
+      val t = a
+      a = n
+      n = t
+      if (a % 4 == 3 && n % 4 == 3) result = -result
+      a %= n
+    }
+    if (n != 1) result = 0
+
+    result
+}
+
+def main(args: Array[String]): Unit =
+{
+    for {
+      a <- 0 until 11
+      n <- 1 until 31 by 2
+    } yield println("n = " + n + ", a = " + a + ": " + jacobi(a, n))
+}

Task/Jacobi-symbol/Scheme/jacobi-symbol.ss

@@ -0,0 +1,13 @@
+(define jacobi (lambda (a n)
+		 (let ((a-mod-n (modulo a n)))
+		   (if (zero? a-mod-n)
+		       (if (= n 1)
+			   1
+			   0)
+		       (if (even? a-mod-n)
+			   (case (modulo n 8)
+			     ((3 5) (- (jacobi (/ a-mod-n 2) n)))
+			     ((1 7) (jacobi (/ a-mod-n 2) n)))
+			   (if (and (= (modulo a-mod-n 4) 3) (= (modulo n 4) 3))
+			       (- (jacobi n a-mod-n))
+			       (jacobi n a-mod-n)))))))

Task/Jacobi-symbol/Sidef/jacobi-symbol.sidef

@@ -0,0 +1,20 @@
+func jacobi(n, k) {
+
+    assert(k > 0,    "#{k} must be positive")
+    assert(k.is_odd, "#{k} must be odd")
+
+    var t = 1
+    while (n %= k) {
+        var v = n.valuation(2)
+        t *= (-1)**v if (k%8 ~~ [3,5])
+        n >>= v
+        (n,k) = (k,n)
+        t = -t if ([n%4, k%4] == [3,3])
+    }
+
+    k==1 ? t : 0
+}
+
+for n in (0..50), k in (0..50) {
+    assert_eq(jacobi(n, 2*k + 1), kronecker(n, 2*k + 1))
+}

Task/Jacobi-symbol/Swift/jacobi-symbol.swift

@@ -0,0 +1,40 @@
+import Foundation
+
+func jacobi(a: Int, n: Int) -> Int {
+  var a = a % n
+  var n = n
+  var res = 1
+
+  while a != 0 {
+    while a & 1 == 0 {
+      a >>= 1
+
+      if n % 8 == 3 || n % 8 == 5 {
+        res = -res
+      }
+    }
+
+    (a, n) = (n, a)
+
+    if a % 4 == 3 && n % 4 == 3 {
+      res = -res
+    }
+
+    a %= n
+  }
+
+  return n == 1 ? res : 0
+}
+
+print("n/a  0  1  2  3  4  5  6  7  8  9")
+print("---------------------------------")
+
+for n in stride(from: 1, through: 17, by: 2) {
+  print(String(format: "%2d", n), terminator: "")
+
+  for a in 0..<10 {
+    print(String(format: " % d", jacobi(a: a, n: n)), terminator: "")
+  }
+
+  print()
+}

Task/Jacobi-symbol/V-(Vlang)/jacobi-symbol.v

@@ -0,0 +1,41 @@
+fn jacobi(aa u64, na u64) ?int {
+    mut a := aa
+    mut n := na
+    if n%2 == 0 {
+        return error("'n' must be a positive odd integer")
+    }
+    a %= n
+    mut result := 1
+    for a != 0 {
+        for a%2 == 0 {
+            a /= 2
+            nn := n % 8
+            if nn == 3 || nn == 5 {
+                result = -result
+            }
+        }
+        a, n = n, a
+        if a%4 == 3 && n%4 == 3 {
+            result = -result
+        }
+        a %= n
+    }
+    if n == 1 {
+        return result
+    }
+    return 0
+}
+
+fn main() {
+    println("Using hand-coded version:")
+    println("n/a  0  1  2  3  4  5  6  7  8  9")
+    println("---------------------------------")
+    for n := u64(1); n <= 17; n += 2 {
+        print("${n:2} ")
+        for a := u64(0); a <= 9; a++ {
+            t := jacobi(a, n)?
+            print(" ${t:2}")
+        }
+        println('')
+    }
+}

Task/Jacobi-symbol/Wren/jacobi-symbol.wren

@@ -0,0 +1,33 @@
+import "/fmt" for Fmt
+
+var jacobi = Fn.new { |a, n|
+    if (!n.isInteger || n <= 0 || n%2 == 0) {
+        Fiber.abort("The 'n' parameter must be an odd positive integer.")
+    }
+    a = a % n
+    var result = 1
+    while (a != 0) {
+        while (a%2  == 0) {
+            a = a / 2
+            var nm8 = n % 8
+            if ([3, 5].contains(nm8)) result = -result
+        }
+        var t = a
+        a = n
+        n = t
+        if (a%4 == 3 && n%4 == 3) result = -result
+        a = a % n
+    }
+    return (n == 1) ? result : 0
+}
+
+System.print("Table of jacobi(a, n):")
+System.print("n/a   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15")
+System.print("---------------------------------------------------------------")
+var n = 1
+while (n < 31) {
+    System.write(Fmt.d(3, n))
+    for (a in 1..15) System.write(Fmt.d(4, jacobi.call(a, n)))
+    System.print()
+    n = n + 2
+}

Task/Jacobi-symbol/XPL0/jacobi-symbol.xpl0

@@ -0,0 +1,39 @@
+func Jacobi(A, N);
+int A, N, Result, T;
+[if A >= N then A:= rem(A/N);
+Result:= 1;
+while A do
+    [while (A&1) = 0 do
+        [A:= A >> 1;
+        if (N&7) = 3 or (N&7) = 5 then Result:= -Result;
+        ];
+    T:= A;  A:= N;  N:= T;
+    if (A&3) = 3 and (N&3) = 3 then Result:= -Result;
+    A:= rem(A/N);
+    ];
+if N = 1 then return Result;
+return 0;
+];
+
+proc PrintTable(KMax, NMax);
+int  KMax, NMax, K, N;
+[Text(0, "N\K|");
+Format(3, 0);
+for K:= 0 to KMax do RlOut(0, float(K));
+CrLf(0);
+Text(0, "----");
+for K:= 0 to KMax do Text(0, "---");
+CrLf(0);
+for N:= 1 to NMax do
+    [Format(2, 0);
+    RlOut(0, float(N));
+    Text(0, " |");
+    Format(3, 0);
+    for K:= 0 to KMax do
+        RlOut(0, float(Jacobi(K, N)));
+    CrLf(0);
+    N:= N+1;
+    ];
+];
+
+PrintTable(20, 21);

Task/Jacobi-symbol/Zkl/jacobi-symbol-1.zkl

@@ -0,0 +1,15 @@
+fcn jacobi(a,n){
+   if(n.isEven or n<1)
+      throw(Exception.ValueError("'n' must be a positive odd integer"));
+   a=a%n;   result,t := 1,0;
+   while(a!=0){
+      while(a.isEven){
+	 a/=2;   n_mod_8:=n%8;
+	 if(n_mod_8==3 or n_mod_8==5) result=-result;
+      }
+      t,a,n = a,n,t;
+      if(a%4==3 and n%4==3) result=-result;
+      a=a%n;
+   }
+   if(n==1) result else 0
+}

Task/Jacobi-symbol/Zkl/jacobi-symbol-2.zkl

@@ -0,0 +1,8 @@
+println("Using hand-coded version:");
+println("n/a  0  1  2  3  4  5  6  7  8  9");
+println("---------------------------------");
+foreach n in ([1..17,2]){
+   print("%2d ".fmt(n));
+   foreach a in (10){ print(" % d".fmt(jacobi(a,n))) }
+   println();
+}

Task/Jacobi-symbol/Zkl/jacobi-symbol-3.zkl

@@ -0,0 +1,9 @@
+var [const] BI=Import.lib("zklBigNum");  // libGMP
+println("\nUsing BigInt library function:");
+println("n/a  0  1  2  3  4  5  6  7  8  9");
+println("---------------------------------");
+foreach n in ([1..17,2]){
+   print("%2d ".fmt(n));
+   foreach a in (10){ print(" % d".fmt(BI(a).jacobi(n))) }
+   println();
+}