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3
Task/Chernicks-Carmichael-numbers/00-META.yaml
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3
Task/Chernicks-Carmichael-numbers/00-META.yaml
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---
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from: http://rosettacode.org/wiki/Chernick's_Carmichael_numbers
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note: Mathematics
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53
Task/Chernicks-Carmichael-numbers/00-TASK.txt
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53
Task/Chernicks-Carmichael-numbers/00-TASK.txt
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[[category:Prime Numbers]]
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In 1939, Jack Chernick proved that, for '''n ≥ 3''' and '''m ≥ 1''':
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U(n, m) = (6m + 1) * (12m + 1) * Product_{i=1..n-2} (2^i * 9m + 1)
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is a [https://en.wikipedia.org/wiki/Carmichael_number Carmichael number] if all the factors are primes and, for '''n > 4''', '''m''' is a multiple of '''2^(n-4)'''.
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;Example
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U(3, m) = (6m + 1) * (12m + 1) * (18m + 1)
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U(4, m) = U(3, m) * (2^2 * 9m + 1)
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U(5, m) = U(4, m) * (2^3 * 9m + 1)
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...
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U(n, m) = U(n-1, m) * (2^(n-2) * 9m + 1)
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* The smallest Chernick's Carmichael number with '''3''' prime factors, is: U(3, 1) = 1729.
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* The smallest Chernick's Carmichael number with '''4''' prime factors, is: U(4, 1) = 63973.
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* The smallest Chernick's Carmichael number with '''5''' prime factors, is: U(5, 380) = 26641259752490421121.
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For '''n = 5''', the smallest number '''m''' that satisfy Chernick's conditions, is '''m = 380''', therefore '''U(5, 380)''' is the smallest Chernick's Carmichael number with '''5''' prime factors.
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'''U(5, 380)''' is a Chernick's Carmichael number because '''m = 380''' is a multiple of '''2^(n-4)''', where '''n = 5''', and the factors { (6*380 + 1), (12*380 + 1), (18*380 + 1), (36*380 + 1), (72*380 + 1) } are all prime numbers.
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;Task
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For '''n ≥ 3''', let '''a(n)''' be the smallest Chernick's Carmichael number with '''n''' prime factors.
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* Compute '''a(n)''' for '''n = 3..9'''.
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* Optional: find '''a(10)'''.
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'''Note''': it's perfectly acceptable to show the terms in factorized form:
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a(3) = 7 * 13 * 19
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a(4) = 7 * 13 * 19 * 37
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a(5) = 2281 * 4561 * 6841 * 13681 * 27361
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...
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;See also
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* [http://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf Jack Chernick, On Fermat's simple theorem (PDF)]
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* [https://oeis.org/A318646 OEIS A318646: The least Chernick's "universal form" Carmichael number with n prime factors]
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; Related tasks
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* [[Carmichael 3 strong pseudoprimes]]
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<br><br>
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#include <gmp.h>
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#include <iostream>
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using namespace std;
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typedef unsigned long long int u64;
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bool primality_pretest(u64 k) { // for k > 23
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if (!(k % 3) || !(k % 5) || !(k % 7) || !(k % 11) ||
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!(k % 13) || !(k % 17) || !(k % 19) || !(k % 23)
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) {
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return (k <= 23);
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}
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return true;
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}
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bool probprime(u64 k, mpz_t n) {
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mpz_set_ui(n, k);
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return mpz_probab_prime_p(n, 0);
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}
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bool is_chernick(int n, u64 m, mpz_t z) {
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if (!primality_pretest(6 * m + 1)) {
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return false;
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}
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if (!primality_pretest(12 * m + 1)) {
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return false;
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}
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u64 t = 9 * m;
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for (int i = 1; i <= n - 2; i++) {
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if (!primality_pretest((t << i) + 1)) {
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return false;
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}
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}
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if (!probprime(6 * m + 1, z)) {
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return false;
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}
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if (!probprime(12 * m + 1, z)) {
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return false;
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}
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for (int i = 1; i <= n - 2; i++) {
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if (!probprime((t << i) + 1, z)) {
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return false;
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}
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}
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return true;
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}
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int main() {
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mpz_t z;
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mpz_inits(z, NULL);
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for (int n = 3; n <= 10; n++) {
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// `m` is a multiple of 2^(n-4), for n > 4
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u64 multiplier = (n > 4) ? (1 << (n - 4)) : 1;
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// For n > 5, m is also a multiple of 5
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if (n > 5) {
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multiplier *= 5;
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}
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for (u64 k = 1; ; k++) {
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u64 m = k * multiplier;
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if (is_chernick(n, m, z)) {
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cout << "a(" << n << ") has m = " << m << endl;
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break;
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}
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}
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}
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return 0;
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}
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#include <stdio.h>
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#include <stdlib.h>
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#include <gmp.h>
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typedef unsigned long long int u64;
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#define TRUE 1
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#define FALSE 0
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int primality_pretest(u64 k) {
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if (!(k % 3) || !(k % 5) || !(k % 7) || !(k % 11) || !(k % 13) || !(k % 17) || !(k % 19) || !(k % 23)) return (k <= 23);
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return TRUE;
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}
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int probprime(u64 k, mpz_t n) {
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mpz_set_ui(n, k);
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return mpz_probab_prime_p(n, 0);
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}
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int is_chernick(int n, u64 m, mpz_t z) {
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u64 t = 9 * m;
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if (primality_pretest(6 * m + 1) == FALSE) return FALSE;
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if (primality_pretest(12 * m + 1) == FALSE) return FALSE;
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for (int i = 1; i <= n - 2; i++) if (primality_pretest((t << i) + 1) == FALSE) return FALSE;
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if (probprime(6 * m + 1, z) == FALSE) return FALSE;
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if (probprime(12 * m + 1, z) == FALSE) return FALSE;
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for (int i = 1; i <= n - 2; i++) if (probprime((t << i) + 1, z) == FALSE) return FALSE;
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return TRUE;
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}
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int main(int argc, char const *argv[]) {
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mpz_t z;
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mpz_inits(z, NULL);
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for (int n = 3; n <= 10; n ++) {
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u64 multiplier = (n > 4) ? (1 << (n - 4)) : 1;
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if (n > 5) multiplier *= 5;
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for (u64 k = 1; ; k++) {
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u64 m = k * multiplier;
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if (is_chernick(n, m, z) == TRUE) {
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printf("a(%d) has m = %llu\n", n, m);
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break;
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}
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}
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}
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return 0;
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}
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@ -0,0 +1,5 @@
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// Generate Chernick's Carmichael numbers. Nigel Galloway: June 1st., 2019
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let fMk m k=isPrime(6*m+1) && isPrime(12*m+1) && [1..k-2]|>List.forall(fun n->isPrime(9*(pown 2 n)*m+1))
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let fX k=Seq.initInfinite(fun n->(n+1)*(pown 2 (k-4))) |> Seq.filter(fun n->fMk n k )
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let cherCar k=let m=Seq.head(fX k) in printfn "m=%d primes -> %A " m ([6*m+1;12*m+1]@List.init(k-2)(fun n->9*(pown 2 (n+1))*m+1))
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[4..9] |> Seq.iter cherCar
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#include "isprime.bas"
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Function PrimalityPretest(k As Integer) As Boolean
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Dim As Integer ppp(1 To 8) = {3,5,7,11,13,17,19,23}
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For i As Integer = 1 To Ubound(ppp)
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If k Mod ppp(i) = 0 Then Return (k <= 23)
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Next i
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Return True
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End Function
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Function isChernick(n As Integer, m As Integer) As Boolean
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Dim As Integer i, t = 9 * m
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If Not PrimalityPretest(6 * m + 1) Then Return False
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If Not PrimalityPretest(12 * m + 1) Then Return False
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For i = 1 To n-1
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If Not PrimalityPretest(t * (2 ^ i) + 1) Then Return False
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Next i
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If Not isPrime(6 * m + 1) Then Return False
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If Not isPrime(12 * m + 1) Then Return False
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For i = 1 To n - 2
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If Not isPrime(t * (2 ^ i) + 1) Then Return False
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Next i
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Return True
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End Function
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Dim As Uinteger multiplier, k, m = 1
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For n As Integer = 3 To 9
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multiplier = Iif (n > 4, 2 ^ (n-4), 1)
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If n > 5 Then multiplier *= 5
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k = 1
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Do
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m = k * multiplier
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If isChernick(n, m) Then
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Print "a(" & n & ") has m = " & m
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Exit Do
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End If
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k += 1
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Loop
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Next n
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Sleep
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package main
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import (
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"fmt"
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"math/big"
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)
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var (
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zero = new(big.Int)
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prod = new(big.Int)
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fact = new(big.Int)
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)
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func ccFactors(n, m uint64) (*big.Int, bool) {
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prod.SetUint64(6*m + 1)
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if !prod.ProbablyPrime(0) {
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return zero, false
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}
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fact.SetUint64(12*m + 1)
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if !fact.ProbablyPrime(0) { // 100% accurate up to 2 ^ 64
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return zero, false
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}
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prod.Mul(prod, fact)
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for i := uint64(1); i <= n-2; i++ {
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fact.SetUint64((1<<i)*9*m + 1)
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if !fact.ProbablyPrime(0) {
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return zero, false
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}
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prod.Mul(prod, fact)
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}
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return prod, true
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}
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func ccNumbers(start, end uint64) {
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for n := start; n <= end; n++ {
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m := uint64(1)
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if n > 4 {
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m = 1 << (n - 4)
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}
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for {
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num, ok := ccFactors(n, m)
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if ok {
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fmt.Printf("a(%d) = %d\n", n, num)
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break
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}
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if n <= 4 {
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m++
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} else {
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m += 1 << (n - 4)
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}
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}
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}
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}
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func main() {
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ccNumbers(3, 9)
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}
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@ -0,0 +1,95 @@
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package main
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import (
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"fmt"
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big "github.com/ncw/gmp"
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)
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const (
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min = 3
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max = 10
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)
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var (
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prod = new(big.Int)
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fact = new(big.Int)
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factors = [max]uint64{}
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bigFactors = [max]*big.Int{}
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)
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func init() {
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for i := 0; i < max; i++ {
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bigFactors[i] = big.NewInt(0)
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}
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}
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func isPrimePretest(k uint64) bool {
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if k%3 == 0 || k%5 == 0 || k%7 == 0 || k%11 == 0 ||
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k%13 == 0 || k%17 == 0 || k%19 == 0 || k%23 == 0 {
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return k <= 23
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}
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return true
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}
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func ccFactors(n, m uint64) bool {
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if !isPrimePretest(6*m + 1) {
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return false
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}
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if !isPrimePretest(12*m + 1) {
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return false
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}
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factors[0] = 6*m + 1
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factors[1] = 12*m + 1
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t := 9 * m
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for i := uint64(1); i <= n-2; i++ {
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tt := (t << i) + 1
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if !isPrimePretest(tt) {
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return false
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}
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factors[i+1] = tt
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}
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for i := 0; i < int(n); i++ {
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fact.SetUint64(factors[i])
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if !fact.ProbablyPrime(0) {
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return false
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}
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bigFactors[i].Set(fact)
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}
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return true
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}
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func prodFactors(n uint64) *big.Int {
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prod.Set(bigFactors[0])
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for i := 1; i < int(n); i++ {
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prod.Mul(prod, bigFactors[i])
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}
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return prod
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}
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func ccNumbers(start, end uint64) {
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for n := start; n <= end; n++ {
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mult := uint64(1)
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if n > 4 {
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mult = 1 << (n - 4)
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}
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if n > 5 {
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mult *= 5
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}
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m := mult
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for {
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if ccFactors(n, m) {
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num := prodFactors(n)
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fmt.Printf("a(%d) = %d\n", n, num)
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fmt.Printf("m(%d) = %d\n", n, m)
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fmt.Println("Factors:", factors[:n], "\n")
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break
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}
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m += mult
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}
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}
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}
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func main() {
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ccNumbers(min, max)
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}
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@ -0,0 +1,10 @@
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a=: {{)v
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if.3=y do.1729 return.end.
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m=. z=. 2^y-4
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f=. 6 12,9*2^}.i.y-1
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while.do.
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uf=.1+f*m
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if.*/1 p: uf do. */x:uf return.end.
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m=.m+z
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end.
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}}
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@ -0,0 +1,14 @@
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a 3
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1729
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a 4
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63973
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a 5
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26641259752490421121
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a 6
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1457836374916028334162241
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a 7
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24541683183872873851606952966798288052977151461406721
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a 8
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53487697914261966820654105730041031613370337776541835775672321
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a 9
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58571442634534443082821160508299574798027946748324125518533225605795841
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@ -0,0 +1,194 @@
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import java.math.BigInteger;
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import java.util.ArrayList;
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import java.util.List;
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public class ChernicksCarmichaelNumbers {
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public static void main(String[] args) {
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for ( long n = 3 ; n < 10 ; n++ ) {
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long m = 0;
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boolean foundComposite = true;
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List<Long> factors = null;
|
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while ( foundComposite ) {
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m += (n <= 4 ? 1 : (long) Math.pow(2, n-4) * 5);
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factors = U(n, m);
|
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foundComposite = false;
|
||||
for ( long factor : factors ) {
|
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if ( ! isPrime(factor) ) {
|
||||
foundComposite = true;
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||||
break;
|
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}
|
||||
}
|
||||
}
|
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System.out.printf("U(%d, %d) = %s = %s %n", n, m, display(factors), multiply(factors));
|
||||
}
|
||||
}
|
||||
|
||||
private static String display(List<Long> factors) {
|
||||
return factors.toString().replace("[", "").replace("]", "").replaceAll(", ", " * ");
|
||||
}
|
||||
|
||||
private static BigInteger multiply(List<Long> factors) {
|
||||
BigInteger result = BigInteger.ONE;
|
||||
for ( long factor : factors ) {
|
||||
result = result.multiply(BigInteger.valueOf(factor));
|
||||
}
|
||||
return result;
|
||||
}
|
||||
|
||||
private static List<Long> U(long n, long m) {
|
||||
List<Long> factors = new ArrayList<>();
|
||||
factors.add(6*m + 1);
|
||||
factors.add(12*m + 1);
|
||||
for ( int i = 1 ; i <= n-2 ; i++ ) {
|
||||
factors.add(((long)Math.pow(2, i)) * 9 * m + 1);
|
||||
}
|
||||
return factors;
|
||||
}
|
||||
|
||||
private static final int MAX = 100_000;
|
||||
private static final boolean[] primes = new boolean[MAX];
|
||||
private static boolean SIEVE_COMPLETE = false;
|
||||
|
||||
private static final boolean isPrimeTrivial(long test) {
|
||||
if ( ! SIEVE_COMPLETE ) {
|
||||
sieve();
|
||||
SIEVE_COMPLETE = true;
|
||||
}
|
||||
return primes[(int) test];
|
||||
}
|
||||
|
||||
private static final void sieve() {
|
||||
// primes
|
||||
for ( int i = 2 ; i < MAX ; i++ ) {
|
||||
primes[i] = true;
|
||||
}
|
||||
for ( int i = 2 ; i < MAX ; i++ ) {
|
||||
if ( primes[i] ) {
|
||||
for ( int j = 2*i ; j < MAX ; j += i ) {
|
||||
primes[j] = false;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// See http://primes.utm.edu/glossary/page.php?sort=StrongPRP
|
||||
public static final boolean isPrime(long testValue) {
|
||||
if ( testValue == 2 ) return true;
|
||||
if ( testValue % 2 == 0 ) return false;
|
||||
if ( testValue <= MAX ) return isPrimeTrivial(testValue);
|
||||
long d = testValue-1;
|
||||
int s = 0;
|
||||
while ( d % 2 == 0 ) {
|
||||
s += 1;
|
||||
d /= 2;
|
||||
}
|
||||
if ( testValue < 1373565L ) {
|
||||
if ( ! aSrp(2, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(3, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
return true;
|
||||
}
|
||||
if ( testValue < 4759123141L ) {
|
||||
if ( ! aSrp(2, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(7, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(61, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
return true;
|
||||
}
|
||||
if ( testValue < 10000000000000000L ) {
|
||||
if ( ! aSrp(3, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(24251, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
return true;
|
||||
}
|
||||
// Try 5 "random" primes
|
||||
if ( ! aSrp(37, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(47, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(61, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(73, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
if ( ! aSrp(83, s, d, testValue) ) {
|
||||
return false;
|
||||
}
|
||||
//throw new RuntimeException("ERROR isPrime: Value too large = "+testValue);
|
||||
return true;
|
||||
}
|
||||
|
||||
private static final boolean aSrp(int a, int s, long d, long n) {
|
||||
long modPow = modPow(a, d, n);
|
||||
//System.out.println("a = "+a+", s = "+s+", d = "+d+", n = "+n+", modpow = "+modPow);
|
||||
if ( modPow == 1 ) {
|
||||
return true;
|
||||
}
|
||||
int twoExpR = 1;
|
||||
for ( int r = 0 ; r < s ; r++ ) {
|
||||
if ( modPow(modPow, twoExpR, n) == n-1 ) {
|
||||
return true;
|
||||
}
|
||||
twoExpR *= 2;
|
||||
}
|
||||
return false;
|
||||
}
|
||||
|
||||
private static final long SQRT = (long) Math.sqrt(Long.MAX_VALUE);
|
||||
|
||||
public static final long modPow(long base, long exponent, long modulus) {
|
||||
long result = 1;
|
||||
while ( exponent > 0 ) {
|
||||
if ( exponent % 2 == 1 ) {
|
||||
if ( result > SQRT || base > SQRT ) {
|
||||
result = multiply(result, base, modulus);
|
||||
}
|
||||
else {
|
||||
result = (result * base) % modulus;
|
||||
}
|
||||
}
|
||||
exponent >>= 1;
|
||||
if ( base > SQRT ) {
|
||||
base = multiply(base, base, modulus);
|
||||
}
|
||||
else {
|
||||
base = (base * base) % modulus;
|
||||
}
|
||||
}
|
||||
return result;
|
||||
}
|
||||
|
||||
|
||||
// Result is a*b % mod, without overflow.
|
||||
public static final long multiply(long a, long b, long modulus) {
|
||||
long x = 0;
|
||||
long y = a % modulus;
|
||||
long t;
|
||||
while ( b > 0 ) {
|
||||
if ( b % 2 == 1 ) {
|
||||
t = x + y;
|
||||
x = (t > modulus ? t-modulus : t);
|
||||
}
|
||||
t = y << 1;
|
||||
y = (t > modulus ? t-modulus : t);
|
||||
b >>= 1;
|
||||
}
|
||||
return x % modulus;
|
||||
}
|
||||
|
||||
}
|
||||
|
|
@ -0,0 +1,108 @@
|
|||
using Primes
|
||||
|
||||
function trial_pretest(k::UInt64)
|
||||
|
||||
if ((k % 3)==0 || (k % 5)==0 || (k % 7)==0 || (k % 11)==0 ||
|
||||
(k % 13)==0 || (k % 17)==0 || (k % 19)==0 || (k % 23)==0)
|
||||
return (k <= 23)
|
||||
end
|
||||
|
||||
return true
|
||||
end
|
||||
|
||||
function gcd_pretest(k::UInt64)
|
||||
|
||||
if (k <= 107)
|
||||
return true
|
||||
end
|
||||
|
||||
gcd(29*31*37*41*43*47*53*59*61*67, k) == 1 &&
|
||||
gcd(71*73*79*83*89*97*101*103*107, k) == 1
|
||||
end
|
||||
|
||||
function is_chernick(n::Int64, m::UInt64)
|
||||
|
||||
t = 9*m
|
||||
|
||||
if (!trial_pretest(6*m + 1))
|
||||
return false
|
||||
end
|
||||
|
||||
if (!trial_pretest(12*m + 1))
|
||||
return false
|
||||
end
|
||||
|
||||
for i in 1:n-2
|
||||
if (!trial_pretest((t << i) + 1))
|
||||
return false
|
||||
end
|
||||
end
|
||||
|
||||
if (!gcd_pretest(6*m + 1))
|
||||
return false
|
||||
end
|
||||
|
||||
if (!gcd_pretest(12*m + 1))
|
||||
return false
|
||||
end
|
||||
|
||||
for i in 1:n-2
|
||||
if (!gcd_pretest((t << i) + 1))
|
||||
return false
|
||||
end
|
||||
end
|
||||
|
||||
if (!isprime(6*m + 1))
|
||||
return false
|
||||
end
|
||||
|
||||
if (!isprime(12*m + 1))
|
||||
return false
|
||||
end
|
||||
|
||||
for i in 1:n-2
|
||||
if (!isprime((t << i) + 1))
|
||||
return false
|
||||
end
|
||||
end
|
||||
|
||||
return true
|
||||
end
|
||||
|
||||
function chernick_carmichael(n::Int64, m::UInt64)
|
||||
prod = big(1)
|
||||
|
||||
prod *= 6*m + 1
|
||||
prod *= 12*m + 1
|
||||
|
||||
for i in 1:n-2
|
||||
prod *= ((big(9)*m)<<i) + 1
|
||||
end
|
||||
|
||||
prod
|
||||
end
|
||||
|
||||
function cc_numbers(from, to)
|
||||
|
||||
for n in from:to
|
||||
|
||||
multiplier = 1
|
||||
|
||||
if (n > 4) multiplier = 1 << (n-4) end
|
||||
if (n > 5) multiplier *= 5 end
|
||||
|
||||
m = UInt64(multiplier)
|
||||
|
||||
while true
|
||||
|
||||
if (is_chernick(n, m))
|
||||
println("a(", n, ") = ", chernick_carmichael(n, m))
|
||||
break
|
||||
end
|
||||
|
||||
m += multiplier
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
cc_numbers(3, 10)
|
||||
|
|
@ -0,0 +1,26 @@
|
|||
ClearAll[PrimeFactorCounts, U]
|
||||
PrimeFactorCounts[n_Integer] := Total[FactorInteger[n][[All, 2]]]
|
||||
U[n_, m_] := (6 m + 1) (12 m + 1) Product[2^i 9 m + 1, {i, 1, n - 2}]
|
||||
FindFirstChernickCarmichaelNumber[n_Integer?Positive] :=
|
||||
Module[{step, i, m, formula, value},
|
||||
step = Ceiling[2^(n - 4)];
|
||||
If[n > 5, step *= 5];
|
||||
i = step;
|
||||
formula = U[n, m];
|
||||
PrintTemporary[Dynamic[i]];
|
||||
While[True,
|
||||
value = formula /. m -> i;
|
||||
If[PrimeFactorCounts[value] == n,
|
||||
Break[];
|
||||
];
|
||||
i += step
|
||||
];
|
||||
{i, value}
|
||||
]
|
||||
FindFirstChernickCarmichaelNumber[3]
|
||||
FindFirstChernickCarmichaelNumber[4]
|
||||
FindFirstChernickCarmichaelNumber[5]
|
||||
FindFirstChernickCarmichaelNumber[6]
|
||||
FindFirstChernickCarmichaelNumber[7]
|
||||
FindFirstChernickCarmichaelNumber[8]
|
||||
FindFirstChernickCarmichaelNumber[9]
|
||||
|
|
@ -0,0 +1,76 @@
|
|||
import strutils, sequtils
|
||||
import bignum
|
||||
|
||||
const
|
||||
Max = 10
|
||||
Factors: array[3..Max, int] = [1, 1, 2, 4, 8, 16, 32, 64] # 1 for n=3 then 2^(n-4).
|
||||
FirstPrimes = [3, 5, 7, 11, 13, 17, 19, 23]
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
iterator factors(n, m: Natural): Natural =
|
||||
## Yield the factors of U(n, m).
|
||||
|
||||
yield 6 * m + 1
|
||||
yield 12 * m + 1
|
||||
var k = 2
|
||||
for _ in 1..(n - 2):
|
||||
yield 9 * k * m + 1
|
||||
inc k, k
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc mayBePrime(n: int): bool =
|
||||
## First primality test.
|
||||
|
||||
if n < 23: return true
|
||||
|
||||
for p in FirstPrimes:
|
||||
if n mod p == 0:
|
||||
return false
|
||||
|
||||
result = true
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc isChernick(n, m: Natural): bool =
|
||||
## Check if U(N, m) if a Chernick-Carmichael number.
|
||||
|
||||
# Use the first and quick test.
|
||||
for factor in factors(n, m):
|
||||
if not factor.mayBePrime():
|
||||
return false
|
||||
|
||||
# Use the slow probability test (need to use a big int).
|
||||
for factor in factors(n, m):
|
||||
if probablyPrime(newInt(factor), 25) == 0:
|
||||
return false
|
||||
|
||||
result = true
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc a(n: Natural): tuple[m: Natural, factors: seq[Natural]] =
|
||||
## For a given "n", find the smallest Charnick-Carmichael number.
|
||||
|
||||
var m: Natural = 0
|
||||
var incr = (if n >= 5: 5 else: 1) * Factors[n] # For n >= 5, a(n) is a multiple of 5.
|
||||
|
||||
while true:
|
||||
inc m, incr
|
||||
if isChernick(n, m):
|
||||
return (m, toSeq(factors(n, m)))
|
||||
|
||||
#———————————————————————————————————————————————————————————————————————————————————————————————————
|
||||
|
||||
import strformat
|
||||
|
||||
for n in 3..Max:
|
||||
let (m, factors) = a(n)
|
||||
|
||||
stdout.write fmt"a({n}) = U({n}, {m}) = "
|
||||
var s = ""
|
||||
for factor in factors:
|
||||
s.addSep(" × ")
|
||||
s.add($factor)
|
||||
stdout.write s, '\n'
|
||||
|
|
@ -0,0 +1,7 @@
|
|||
cherCar(n)={
|
||||
my(C=vector(n));C[1]=6; C[2]=12; for(g=3,n,C[g]=2^(g-2)*9);
|
||||
my(i=1); my(N(g)=while(i<=n&ispseudoprime(g*C[i]+1),i=i+1); return(i>n));
|
||||
i=1; my(G(g)=while(i<=n&isprime(g*C[i]+1),i=i+1); return(i>n));
|
||||
i=1; if(n>4,i=2^(n-4)); if(n>5,i=i*5); my(m=i); while(!(N(m)&G(m)),m=m+i);
|
||||
printf("cherCar(%d): m = %d\n",n,m)}
|
||||
for(x=3,9,cherCar(x))
|
||||
|
|
@ -0,0 +1,24 @@
|
|||
use 5.020;
|
||||
use warnings;
|
||||
use ntheory qw/:all/;
|
||||
use experimental qw/signatures/;
|
||||
|
||||
sub chernick_carmichael_factors ($n, $m) {
|
||||
(6*$m + 1, 12*$m + 1, (map { (1 << $_) * 9*$m + 1 } 1 .. $n-2));
|
||||
}
|
||||
|
||||
sub chernick_carmichael_number ($n, $callback) {
|
||||
|
||||
my $multiplier = ($n > 4) ? (1 << ($n-4)) : 1;
|
||||
|
||||
for (my $m = 1 ; ; ++$m) {
|
||||
my @f = chernick_carmichael_factors($n, $m * $multiplier);
|
||||
next if not vecall { is_prime($_) } @f;
|
||||
$callback->(@f);
|
||||
last;
|
||||
}
|
||||
}
|
||||
|
||||
foreach my $n (3..9) {
|
||||
chernick_carmichael_number($n, sub (@f) { say "a($n) = ", vecprod(@f) });
|
||||
}
|
||||
|
|
@ -0,0 +1,40 @@
|
|||
(phixonline)-->
|
||||
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">chernick_carmichael_factors</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">6</span><span style="color: #0000FF;">*</span><span style="color: #000000;">m</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">12</span><span style="color: #0000FF;">*</span><span style="color: #000000;">m</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;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">power</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: #0000FF;">*</span> <span style="color: #000000;">9</span><span style="color: #0000FF;">*</span><span style="color: #000000;">m</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</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;">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;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">m_prime</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: #7060A8;">mpz_set_d</span><span style="color: #0000FF;">(</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: #008080;">return</span> <span style="color: #7060A8;">mpz_prime</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;">function</span>
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">is_chernick_carmichael</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">)</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;">2</span> <span style="color: #0000FF;">?</span> <span style="color: #000000;">m_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">6</span><span style="color: #0000FF;">*</span><span style="color: #000000;">m</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">m_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">12</span><span style="color: #0000FF;">*</span><span style="color: #000000;">m</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: #000000;">m_prime</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</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: #0000FF;">*</span> <span style="color: #000000;">9</span><span style="color: #0000FF;">*</span><span style="color: #000000;">m</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span>
|
||||
<span style="color: #000000;">is_chernick_carmichael</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;">m</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;">chernick_carmichael_number</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</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;">4</span> <span style="color: #0000FF;">?</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</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: #0000FF;">:</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">mm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span>
|
||||
<span style="color: #008080;">while</span> <span style="color: #008080;">not</span> <span style="color: #000000;">is_chernick_carmichael</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">mm</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> <span style="color: #000000;">mm</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">m</span> <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">chernick_carmichael_factors</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">mm</span><span style="color: #0000FF;">),</span><span style="color: #000000;">mm</span><span style="color: #0000FF;">}</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">3</span> <span style="color: #008080;">to</span> <span style="color: #000000;">9</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">chernick_carmichael_number</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpz_set_si</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: #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;">f</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #7060A8;">mpz_mul_d</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;">f</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
|
||||
<span style="color: #000000;">f</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: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</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;">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;">"U(%d,%d): %s = %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">),</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" * "</span><span style="color: #0000FF;">)})</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<!--
|
||||
|
|
@ -0,0 +1,53 @@
|
|||
(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;">sequence</span> <span style="color: #000000;">ppp</span> <span style="color: #0000FF;">=</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: #008080;">function</span> <span style="color: #000000;">primality_pretest</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</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: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ppp</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ppp</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</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: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">23</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>
|
||||
<span style="color: #008080;">return</span> <span style="color: #004600;">true</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">probprime</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">mpz</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpz_set_d</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #7060A8;">mpz_prime</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;">function</span>
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">is_chernick</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</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: #004080;">mpz</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">9</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">;</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">primality_pretest</span><span style="color: #0000FF;">(</span><span style="color: #000000;">6</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">m</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: #004600;">false</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;">if</span> <span style="color: #000000;">primality_pretest</span><span style="color: #0000FF;">(</span><span style="color: #000000;">12</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">m</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: #004600;">false</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;">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;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">3</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">primality_pretest</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">power</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: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">==</span> <span style="color: #004600;">false</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;">if</span> <span style="color: #000000;">probprime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">6</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">==</span> <span style="color: #004600;">false</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;">if</span> <span style="color: #000000;">probprime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">12</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">==</span> <span style="color: #004600;">false</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;">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;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">probprime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">power</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: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">==</span> <span style="color: #004600;">false</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: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #008080;">procedure</span> <span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
|
||||
<span style="color: #004080;">mpz</span> <span style="color: #000000;">z</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</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;">3</span> <span style="color: #008080;">to</span> <span style="color: #000000;">10</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">multiplier</span> <span style="color: #0000FF;">=</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;">4</span> <span style="color: #0000FF;">?</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</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: #0000FF;">:</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">></span><span style="color: #000000;">5</span> <span style="color: #008080;">then</span> <span style="color: #000000;">multiplier</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">5</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
|
||||
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">10</span> <span style="color: #008080;">then</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">12564168</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- cheat!</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">multiplier</span><span style="color: #0000FF;">;</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">is_chernick</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">z</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;">"a(%d) has m = %d\n"</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;">m</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: #000000;">k</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;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #0000FF;">?</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
|
||||
<span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
|
||||
<!--
|
||||
|
|
@ -0,0 +1,29 @@
|
|||
?- use_module(library(primality)).
|
||||
|
||||
u(3, M, A * B * C) :-
|
||||
A is 6*M + 1, B is 12*M + 1, C is 18*M + 1, !.
|
||||
u(N, M, U0 * D) :-
|
||||
succ(Pn, N), u(Pn, M, U0),
|
||||
D is 9*(1 << (N - 2))*M + 1.
|
||||
|
||||
prime_factorization(A*B) :- prime(B), prime_factorization(A), !.
|
||||
prime_factorization(A) :- prime(A).
|
||||
|
||||
step(N, 1) :- N < 5, !.
|
||||
step(5, 2) :- !.
|
||||
step(N, K) :- K is 5*(1 << (N - 4)).
|
||||
|
||||
a(N, Factors) :- % due to backtracking nature of Prolog, a(n) will return all chernick-carmichael numbers.
|
||||
N > 2, !,
|
||||
step(N, I),
|
||||
between(1, infinite, J), M is I * J,
|
||||
u(N, M, Factors),
|
||||
prime_factorization(Factors).
|
||||
|
||||
main :-
|
||||
forall(
|
||||
(between(3, 9, K), once(a(K, Factorization)), N is Factorization),
|
||||
format("~w: ~w = ~w~n", [K, Factorization, N])),
|
||||
halt.
|
||||
|
||||
?- main.
|
||||
|
|
@ -0,0 +1,45 @@
|
|||
prime(N) :-
|
||||
integer(N),
|
||||
N > 1,
|
||||
divcheck(
|
||||
N,
|
||||
[ 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],
|
||||
Result),
|
||||
((Result = prime, !); miller_rabin_primality_test(N)).
|
||||
|
||||
divcheck(_, [], unknown) :- !.
|
||||
divcheck(N, [P|_], prime) :- P*P > N, !.
|
||||
divcheck(N, [P|Ps], State) :- N mod P =\= 0, divcheck(N, Ps, State).
|
||||
|
||||
miller_rabin_primality_test(N) :-
|
||||
bases(Bases, N),
|
||||
forall(member(A, Bases), strong_fermat_pseudoprime(N, A)).
|
||||
|
||||
miller_rabin_precision(16).
|
||||
|
||||
bases([31, 73], N) :- N < 9_080_191, !.
|
||||
bases([2, 7, 61], N) :- N < 4_759_123_141, !.
|
||||
bases([2, 325, 9_375, 28_178, 450_775, 9_780_504, 1_795_265_022], N) :-
|
||||
N < 18_446_744_073_709_551_616, !. % 2^64
|
||||
bases(Bases, N) :-
|
||||
miller_rabin_precision(T), RndLimit is N - 2,
|
||||
length(Bases, T), maplist(random_between(2, RndLimit), Bases).
|
||||
|
||||
strong_fermat_pseudoprime(N, A) :- % miller-rabin strong pseudoprime test with base A.
|
||||
succ(Pn, N), factor_2s(Pn, S, D),
|
||||
X is powm(A, D, N),
|
||||
((X =:= 1, !); \+ composite_witness(N, S, X)).
|
||||
|
||||
composite_witness(_, 0, _) :- !.
|
||||
composite_witness(N, K, X) :-
|
||||
X =\= N-1,
|
||||
succ(Pk, K), X2 is (X*X) mod N, composite_witness(N, Pk, X2).
|
||||
|
||||
factor_2s(N, S, D) :- factor_2s(0, N, S, D).
|
||||
factor_2s(S, D, S, D) :- D /\ 1 =\= 0, !.
|
||||
factor_2s(S0, D0, S, D) :-
|
||||
succ(S0, S1), D1 is D0 >> 1,
|
||||
factor_2s(S1, D1, S, D).
|
||||
|
|
@ -0,0 +1,66 @@
|
|||
"""
|
||||
|
||||
Python implementation of
|
||||
http://rosettacode.org/wiki/Chernick%27s_Carmichael_numbers
|
||||
|
||||
"""
|
||||
|
||||
# use sympy for prime test
|
||||
|
||||
from sympy import isprime
|
||||
|
||||
# based on C version
|
||||
|
||||
def primality_pretest(k):
|
||||
if not (k % 3) or not (k % 5) or not (k % 7) or not (k % 11) or not(k % 13) or not (k % 17) or not (k % 19) or not (k % 23):
|
||||
return (k <= 23)
|
||||
|
||||
return True
|
||||
|
||||
def is_chernick(n, m):
|
||||
|
||||
t = 9 * m
|
||||
|
||||
if not primality_pretest(6 * m + 1):
|
||||
return False
|
||||
|
||||
if not primality_pretest(12 * m + 1):
|
||||
return False
|
||||
|
||||
for i in range(1,n-1):
|
||||
if not primality_pretest((t << i) + 1):
|
||||
return False
|
||||
|
||||
if not isprime(6 * m + 1):
|
||||
return False
|
||||
|
||||
if not isprime(12 * m + 1):
|
||||
return False
|
||||
|
||||
for i in range(1,n - 1):
|
||||
if not isprime((t << i) + 1):
|
||||
return False
|
||||
|
||||
return True
|
||||
|
||||
for n in range(3,10):
|
||||
|
||||
if n > 4:
|
||||
multiplier = 1 << (n - 4)
|
||||
else:
|
||||
multiplier = 1
|
||||
|
||||
if n > 5:
|
||||
multiplier *= 5
|
||||
|
||||
|
||||
k = 1
|
||||
|
||||
while True:
|
||||
m = k * multiplier
|
||||
|
||||
if is_chernick(n, m):
|
||||
print("a("+str(n)+") has m = "+str(m))
|
||||
break
|
||||
|
||||
k += 1
|
||||
|
|
@ -0,0 +1,23 @@
|
|||
use Inline::Perl5;
|
||||
use ntheory:from<Perl5> <:all>;
|
||||
|
||||
sub chernick-factors ($n, $m) {
|
||||
6×$m + 1, 12×$m + 1, |((1 .. $n-2).map: { (1 +< $_) × 9×$m + 1 } )
|
||||
}
|
||||
|
||||
sub chernick-carmichael-number ($n) {
|
||||
|
||||
my $multiplier = 1 +< (($n-4) max 0);
|
||||
my $iterator = $n < 5 ?? (1 .. *) !! (1 .. *).map: * × 5;
|
||||
|
||||
$multiplier × $iterator.first: -> $m {
|
||||
[&&] chernick-factors($n, $m × $multiplier).map: { is_prime($_) }
|
||||
}
|
||||
|
||||
}
|
||||
|
||||
for 3 .. 9 -> $n {
|
||||
my $m = chernick-carmichael-number($n);
|
||||
my @f = chernick-factors($n, $m);
|
||||
say "U($n, $m): {[×] @f} = {@f.join(' ⨉ ')}";
|
||||
}
|
||||
|
|
@ -0,0 +1,19 @@
|
|||
func chernick_carmichael_factors (n, m) {
|
||||
[6*m + 1, 12*m + 1, {|i| 2**i * 9*m + 1 }.map(1 .. n-2)...]
|
||||
}
|
||||
|
||||
func is_chernick_carmichael (n, m) {
|
||||
(n == 2) ? (is_prime(6*m + 1) && is_prime(12*m + 1))
|
||||
: (is_prime(2**(n-2) * 9*m + 1) && __FUNC__(n-1, m))
|
||||
}
|
||||
|
||||
func chernick_carmichael_number(n, callback) {
|
||||
var multiplier = (n>4 ? 2**(n-4) : 1)
|
||||
var m = (1..Inf -> first {|m| is_chernick_carmichael(n, m * multiplier) })
|
||||
var f = chernick_carmichael_factors(n, m * multiplier)
|
||||
callback(f...)
|
||||
}
|
||||
|
||||
for n in (3..9) {
|
||||
chernick_carmichael_number(n, {|*f| say "a(#{n}) = #{f.join(' * ')}" })
|
||||
}
|
||||
|
|
@ -0,0 +1,62 @@
|
|||
import "/big" for BigInt, BigInts
|
||||
import "/fmt" for Fmt
|
||||
|
||||
var min = 3
|
||||
var max = 9
|
||||
var prod = BigInt.zero
|
||||
var fact = BigInt.zero
|
||||
var factors = List.filled(max, 0)
|
||||
var bigFactors = List.filled(max, null)
|
||||
|
||||
var init = Fn.new {
|
||||
for (i in 0...max) bigFactors[i] = BigInt.zero
|
||||
}
|
||||
|
||||
var isPrimePretest = Fn.new { |k|
|
||||
if (k%3 == 0 || k%5 == 0 || k%7 == 0 || k%11 == 0 ||
|
||||
(k%13 == 0) || k%17 == 0 || k%19 == 0 || k%23 == 0) return k <= 23
|
||||
return true
|
||||
}
|
||||
|
||||
var ccFactors = Fn.new { |n, m|
|
||||
if (!isPrimePretest.call(6*m + 1)) return false
|
||||
if (!isPrimePretest.call(12*m + 1)) return false
|
||||
factors[0] = 6*m + 1
|
||||
factors[1] = 12*m + 1
|
||||
var t = 9 * m
|
||||
var i = 1
|
||||
while (i <= n-2) {
|
||||
var tt = (t << i) + 1
|
||||
if (!isPrimePretest.call(tt)) return false
|
||||
factors[i+1] = tt
|
||||
i = i + 1
|
||||
}
|
||||
for (i in 0...n) {
|
||||
fact = BigInt.new(factors[i])
|
||||
if (!fact.isProbablePrime(1)) return false
|
||||
bigFactors[i] = fact
|
||||
}
|
||||
return true
|
||||
}
|
||||
|
||||
var ccNumbers = Fn.new { |start, end|
|
||||
for (n in start..end) {
|
||||
var mult = 1
|
||||
if (n > 4) mult = 1 << (n - 4)
|
||||
if (n > 5) mult = mult * 5
|
||||
var m = mult
|
||||
while (true) {
|
||||
if (ccFactors.call(n, m)) {
|
||||
var num = BigInts.prod(bigFactors.take(n))
|
||||
Fmt.print("a($d) = $i", n, num)
|
||||
Fmt.print("m($d) = $d", n, m)
|
||||
Fmt.print("Factors: $n\n", factors[0...n])
|
||||
break
|
||||
}
|
||||
m = m + mult
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
init.call()
|
||||
ccNumbers.call(min, max)
|
||||
|
|
@ -0,0 +1,28 @@
|
|||
var [const] BI=Import("zklBigNum"); // libGMP
|
||||
|
||||
fcn ccFactors(n,m){ // not re-entrant
|
||||
prod:=BI(6*m + 1);
|
||||
if(not prod.probablyPrime()) return(False);
|
||||
fact:=BI(12*m + 1);
|
||||
if(not fact.probablyPrime()) return(False);
|
||||
prod.mul(fact);
|
||||
foreach i in ([1..n-2]){
|
||||
fact.set((2).pow(i) *9*m + 1);
|
||||
if(not fact.probablyPrime()) return(False);
|
||||
prod.mul(fact);
|
||||
}
|
||||
prod
|
||||
}
|
||||
|
||||
fcn ccNumbers(start,end){
|
||||
foreach n in ([start..end]){
|
||||
a,m := ( if(n<=4) 1 else (2).pow(n - 4) ), a;
|
||||
while(1){
|
||||
if(num := ccFactors(n,m)){
|
||||
println("a(%d) = %,d".fmt(n,num));
|
||||
break;
|
||||
}
|
||||
m+=a;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1 @@
|
|||
ccNumbers(3,9);
|
||||
Loading…
Add table
Add a link
Reference in a new issue