Chernick's Carmichael numbers
In 1939, Jack Chernick proved that, for n ≥ 3 and m ≥ 1:
You are encouraged to solve this task according to the task description, using any language you may know.
U(n, m) = (6m + 1) * (12m + 1) * Product_{i=1..n-2} (2^i * 9m + 1)
is a Carmichael number if all the factors are primes and, for n > 4, m is a multiple of 2^(n-4).
- Example
U(3, m) = (6m + 1) * (12m + 1) * (18m + 1) U(4, m) = U(3, m) * (2^2 * 9m + 1) U(5, m) = U(4, m) * (2^3 * 9m + 1) ... U(n, m) = U(n-1, m) * (2^(n-2) * 9m + 1)
- The smallest Chernick's Carmichael number with 3 prime factors, is: U(3, 1) = 1729.
- The smallest Chernick's Carmichael number with 4 prime factors, is: U(4, 1) = 63973.
- The smallest Chernick's Carmichael number with 5 prime factors, is: U(5, 380) = 26641259752490421121.
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.
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.
- Task
For n ≥ 3, let a(n) be the smallest Chernick's Carmichael number with n prime factors.
- Compute a(n) for n = 3..9.
- Optional: find a(10).
Note: it's perfectly acceptable to show the terms in factorized form:
a(3) = 7 * 13 * 19 a(4) = 7 * 13 * 19 * 37 a(5) = 2281 * 4561 * 6841 * 13681 * 27361 ...
- See also
- Related tasks
C
#include <stdio.h>
#include <stdlib.h>
#include <gmp.h>
typedef unsigned long long int u64;
#define TRUE 1
#define FALSE 0
int primality_pretest(u64 k) {
if (!(k % 3) || !(k % 5) || !(k % 7) || !(k % 11) || !(k % 13) || !(k % 17) || !(k % 19) || !(k % 23)) return (k <= 23);
return TRUE;
}
int probprime(u64 k, mpz_t n) {
mpz_set_ui(n, k);
return mpz_probab_prime_p(n, 0);
}
int is_chernick(int n, u64 m, mpz_t z) {
u64 t = 9 * m;
if (primality_pretest(6 * m + 1) == FALSE) return FALSE;
if (primality_pretest(12 * m + 1) == FALSE) return FALSE;
for (int i = 1; i <= n - 2; i++) if (primality_pretest((t << i) + 1) == FALSE) return FALSE;
if (probprime(6 * m + 1, z) == FALSE) return FALSE;
if (probprime(12 * m + 1, z) == FALSE) return FALSE;
for (int i = 1; i <= n - 2; i++) if (probprime((t << i) + 1, z) == FALSE) return FALSE;
return TRUE;
}
int main(int argc, char const *argv[]) {
mpz_t z;
mpz_inits(z, NULL);
for (int n = 3; n <= 10; n ++) {
u64 multiplier = (n > 4) ? (1 << (n - 4)) : 1;
if (n > 5) multiplier *= 5;
for (u64 k = 1; ; k++) {
u64 m = k * multiplier;
if (is_chernick(n, m, z) == TRUE) {
printf("a(%d) has m = %llu\n", n, m);
break;
}
}
}
return 0;
}
- Output:
a(3) has m = 1 a(4) has m = 1 a(5) has m = 380 a(6) has m = 380 a(7) has m = 780320 a(8) has m = 950560 a(9) has m = 950560 a(10) has m = 3208386195840
C++
#include <gmp.h>
#include <iostream>
using namespace std;
typedef unsigned long long int u64;
bool primality_pretest(u64 k) { // for k > 23
if (!(k % 3) || !(k % 5) || !(k % 7) || !(k % 11) ||
!(k % 13) || !(k % 17) || !(k % 19) || !(k % 23)
) {
return (k <= 23);
}
return true;
}
bool probprime(u64 k, mpz_t n) {
mpz_set_ui(n, k);
return mpz_probab_prime_p(n, 0);
}
bool is_chernick(int n, u64 m, mpz_t z) {
if (!primality_pretest(6 * m + 1)) {
return false;
}
if (!primality_pretest(12 * m + 1)) {
return false;
}
u64 t = 9 * m;
for (int i = 1; i <= n - 2; i++) {
if (!primality_pretest((t << i) + 1)) {
return false;
}
}
if (!probprime(6 * m + 1, z)) {
return false;
}
if (!probprime(12 * m + 1, z)) {
return false;
}
for (int i = 1; i <= n - 2; i++) {
if (!probprime((t << i) + 1, z)) {
return false;
}
}
return true;
}
int main() {
mpz_t z;
mpz_inits(z, NULL);
for (int n = 3; n <= 10; n++) {
// `m` is a multiple of 2^(n-4), for n > 4
u64 multiplier = (n > 4) ? (1 << (n - 4)) : 1;
// For n > 5, m is also a multiple of 5
if (n > 5) {
multiplier *= 5;
}
for (u64 k = 1; ; k++) {
u64 m = k * multiplier;
if (is_chernick(n, m, z)) {
cout << "a(" << n << ") has m = " << m << endl;
break;
}
}
}
return 0;
}
- Output:
a(3) has m = 1 a(4) has m = 1 a(5) has m = 380 a(6) has m = 380 a(7) has m = 780320 a(8) has m = 950560 a(9) has m = 950560 a(10) has m = 3208386195840
(takes ~3.5 minutes)
F#
// Generate Chernick's Carmichael numbers. Nigel Galloway: June 1st., 2019
open Open.Numeric.Primes
let fMk m k=Number.IsPrime(6UL*m+1UL) && Number.IsPrime(12UL*m+1UL) && [1..k-2]|>List.forall(fun n->Number.IsPrime(9UL*(pown 2UL n)*m+1UL))
let fX k=Seq.initInfinite(fun n->match k with 3->uint64(n+1) |_->uint64(n+1)*(pown 2UL (k-4)))|>Seq.filter(fun n->fMk n k)
let cherCar k=let m=Seq.head(fX k) in sprintf "m=%d primes-> %A " m ([6UL*m+1UL;12UL*m+1UL]@List.init(k-2)(fun n->9UL*(pown 2UL (n+1))*m+1UL))
[3..9] |> Seq.iter(fun g->printfn $"cherCar %d{g}: %s{cherCar g}")
- Output:
cherCar 3: m=1 primes-> [7; 13; 19] cherCar 4: m=1 primes-> [7; 13; 19; 37] cherCar 5: m=380 primes-> [2281; 4561; 6841; 13681; 27361] cherCar 6: m=380 primes-> [2281; 4561; 6841; 13681; 27361; 54721] cherCar 7: m=780320 primes-> [4681921; 9363841; 14045761; 28091521; 56183041; 112366081; 224732161] cherCar 8: m=950560 primes-> [5703361; 11406721; 17110081; 34220161; 68440321; 136880641; 273761281; 547522561] cherCar 9: m=950560 primes-> [5703361; 11406721; 17110081; 34220161; 68440321; 136880641; 273761281; 547522561; 1095045121]
FreeBASIC
Basic only
#include "isprime.bas"
Function PrimalityPretest(k As Integer) As Boolean
Dim As Integer ppp(1 To 8) = {3,5,7,11,13,17,19,23}
For i As Integer = 1 To Ubound(ppp)
If k Mod ppp(i) = 0 Then Return (k <= 23)
Next i
Return True
End Function
Function isChernick(n As Integer, m As Integer) As Boolean
Dim As Integer i, t = 9 * m
If Not PrimalityPretest(6 * m + 1) Then Return False
If Not PrimalityPretest(12 * m + 1) Then Return False
For i = 1 To n-1
If Not PrimalityPretest(t * (2 ^ i) + 1) Then Return False
Next i
If Not isPrime(6 * m + 1) Then Return False
If Not isPrime(12 * m + 1) Then Return False
For i = 1 To n - 2
If Not isPrime(t * (2 ^ i) + 1) Then Return False
Next i
Return True
End Function
Dim As Uinteger multiplier, k, m = 1
For n As Integer = 3 To 9
multiplier = Iif (n > 4, 2 ^ (n-4), 1)
If n > 5 Then multiplier *= 5
k = 1
Do
m = k * multiplier
If isChernick(n, m) Then
Print "a(" & n & ") has m = " & m
Exit Do
End If
k += 1
Loop
Next n
Sleep
Go
Basic only
package main
import (
"fmt"
"math/big"
)
var (
zero = new(big.Int)
prod = new(big.Int)
fact = new(big.Int)
)
func ccFactors(n, m uint64) (*big.Int, bool) {
prod.SetUint64(6*m + 1)
if !prod.ProbablyPrime(0) {
return zero, false
}
fact.SetUint64(12*m + 1)
if !fact.ProbablyPrime(0) { // 100% accurate up to 2 ^ 64
return zero, false
}
prod.Mul(prod, fact)
for i := uint64(1); i <= n-2; i++ {
fact.SetUint64((1<<i)*9*m + 1)
if !fact.ProbablyPrime(0) {
return zero, false
}
prod.Mul(prod, fact)
}
return prod, true
}
func ccNumbers(start, end uint64) {
for n := start; n <= end; n++ {
m := uint64(1)
if n > 4 {
m = 1 << (n - 4)
}
for {
num, ok := ccFactors(n, m)
if ok {
fmt.Printf("a(%d) = %d\n", n, num)
break
}
if n <= 4 {
m++
} else {
m += 1 << (n - 4)
}
}
}
}
func main() {
ccNumbers(3, 9)
}
- Output:
a(3) = 1729 a(4) = 63973 a(5) = 26641259752490421121 a(6) = 1457836374916028334162241 a(7) = 24541683183872873851606952966798288052977151461406721 a(8) = 53487697914261966820654105730041031613370337776541835775672321 a(9) = 58571442634534443082821160508299574798027946748324125518533225605795841
Basic plus optional
To reach a(10) in a reasonable time, a much more efficient approach is needed.
The following version takes account of the optimizations referred to in the Talk page and previewed in the C++ entry above.
It also uses a wrapper for the C library, GMP, which despite the overhead of cgo is still much faster than Go's native big.Int library.
The resulting executable is several hundred times faster than before and, even on my modest Celeron @1.6GHZ, reaches a(9) in under 10ms and a(10) in about 22 minutes.
package main
import (
"fmt"
big "github.com/ncw/gmp"
)
const (
min = 3
max = 10
)
var (
prod = new(big.Int)
fact = new(big.Int)
factors = [max]uint64{}
bigFactors = [max]*big.Int{}
)
func init() {
for i := 0; i < max; i++ {
bigFactors[i] = big.NewInt(0)
}
}
func isPrimePretest(k uint64) bool {
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
}
func ccFactors(n, m uint64) bool {
if !isPrimePretest(6*m + 1) {
return false
}
if !isPrimePretest(12*m + 1) {
return false
}
factors[0] = 6*m + 1
factors[1] = 12*m + 1
t := 9 * m
for i := uint64(1); i <= n-2; i++ {
tt := (t << i) + 1
if !isPrimePretest(tt) {
return false
}
factors[i+1] = tt
}
for i := 0; i < int(n); i++ {
fact.SetUint64(factors[i])
if !fact.ProbablyPrime(0) {
return false
}
bigFactors[i].Set(fact)
}
return true
}
func prodFactors(n uint64) *big.Int {
prod.Set(bigFactors[0])
for i := 1; i < int(n); i++ {
prod.Mul(prod, bigFactors[i])
}
return prod
}
func ccNumbers(start, end uint64) {
for n := start; n <= end; n++ {
mult := uint64(1)
if n > 4 {
mult = 1 << (n - 4)
}
if n > 5 {
mult *= 5
}
m := mult
for {
if ccFactors(n, m) {
num := prodFactors(n)
fmt.Printf("a(%d) = %d\n", n, num)
fmt.Printf("m(%d) = %d\n", n, m)
fmt.Println("Factors:", factors[:n], "\n")
break
}
m += mult
}
}
}
func main() {
ccNumbers(min, max)
}
- Output:
a(3) = 1729 m(3) = 1 Factors: [7 13 19] a(4) = 63973 m(4) = 1 Factors: [7 13 19 37] a(5) = 26641259752490421121 m(5) = 380 Factors: [2281 4561 6841 13681 27361] a(6) = 1457836374916028334162241 m(6) = 380 Factors: [2281 4561 6841 13681 27361 54721] a(7) = 24541683183872873851606952966798288052977151461406721 m(7) = 780320 Factors: [4681921 9363841 14045761 28091521 56183041 112366081 224732161] a(8) = 53487697914261966820654105730041031613370337776541835775672321 m(8) = 950560 Factors: [5703361 11406721 17110081 34220161 68440321 136880641 273761281 547522561] a(9) = 58571442634534443082821160508299574798027946748324125518533225605795841 m(9) = 950560 Factors: [5703361 11406721 17110081 34220161 68440321 136880641 273761281 547522561 1095045121] a(10) = 24616075028246330441656912428380582403261346369700917629170235674289719437963233744091978433592331048416482649086961226304033068172880278517841921 m(10) = 3208386195840 Factors: [19250317175041 38500634350081 57750951525121 115501903050241 231003806100481 462007612200961 924015224401921 1848030448803841 3696060897607681 7392121795215361]
J
Brute force:
a=: {{)v
if.3=y do.1729 return.end.
m=. z=. 2^y-4
f=. 6 12,9*2^}.i.y-1
while.do.
uf=.1+f*m
if.*/1 p: uf do. */x:uf return.end.
m=.m+z
end.
}}
Task examples:
a 3
1729
a 4
63973
a 5
26641259752490421121
a 6
1457836374916028334162241
a 7
24541683183872873851606952966798288052977151461406721
a 8
53487697914261966820654105730041031613370337776541835775672321
a 9
58571442634534443082821160508299574798027946748324125518533225605795841
Java
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;
public class ChernicksCarmichaelNumbers {
public static void main(String[] args) {
for ( long n = 3 ; n < 10 ; n++ ) {
long m = 0;
boolean foundComposite = true;
List<Long> factors = null;
while ( foundComposite ) {
m += (n <= 4 ? 1 : (long) Math.pow(2, n-4) * 5);
factors = U(n, m);
foundComposite = false;
for ( long factor : factors ) {
if ( ! isPrime(factor) ) {
foundComposite = true;
break;
}
}
}
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;
}
}
- Output:
U(3, 1) = 7 * 13 * 19 = 1729 U(4, 1) = 7 * 13 * 19 * 37 = 63973 U(5, 380) = 2281 * 4561 * 6841 * 13681 * 27361 = 26641259752490421121 U(6, 380) = 2281 * 4561 * 6841 * 13681 * 27361 * 54721 = 1457836374916028334162241 U(7, 780320) = 4681921 * 9363841 * 14045761 * 28091521 * 56183041 * 112366081 * 224732161 = 24541683183872873851606952966798288052977151461406721 U(8, 950560) = 5703361 * 11406721 * 17110081 * 34220161 * 68440321 * 136880641 * 273761281 * 547522561 = 53487697914261966820654105730041031613370337776541835775672321 U(9, 950560) = 5703361 * 11406721 * 17110081 * 34220161 * 68440321 * 136880641 * 273761281 * 547522561 * 1095045121 = 58571442634534443082821160508299574798027946748324125518533225605795841
Julia
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)
- Output:
a(3) = 1729 a(4) = 63973 a(5) = 26641259752490421121 a(6) = 1457836374916028334162241 a(7) = 24541683183872873851606952966798288052977151461406721 a(8) = 53487697914261966820654105730041031613370337776541835775672321 a(9) = 58571442634534443082821160508299574798027946748324125518533225605795841 a(10) = 24616075028246330441656912428380582403261346369700917629170235674289719437963233744091978433592331048416482649086961226304033068172880278517841921
(takes ~6.5 minutes)
Mathematica / Wolfram Language
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]
- Output:
{1,1729}
{1,63973}
{380,26641259752490421121}
{380,1457836374916028334162241}
{780320,24541683183872873851606952966798288052977151461406721}
{950560,53487697914261966820654105730041031613370337776541835775672321}
{950560,58571442634534443082821160508299574798027946748324125518533225605795841}
Nim
Until a(9) a simple primality test using divisions by odd numbers is sufficient. But for a(10), it is necessary to improve the test. We have used here some optimizations found in other solutions:
- – eliminating multiples of 3, 5, 7, 11, 13, 17, 19, 23;
- – using a probability test which implies to use big integers; so, we have to convert the tested number to a big integer;
- – for n >= 5, checking only values of m which are multiple of 5 (in fact, we check only the multiples of 5 × 2^(n-4).
With these optimizations, the program executes in 4-5 minutes.
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'
- Output:
a(3) = U(3, 1) = 7 × 13 × 19 a(4) = U(4, 1) = 7 × 13 × 19 × 37 a(5) = U(5, 380) = 2281 × 4561 × 6841 × 13681 × 27361 a(6) = U(6, 380) = 2281 × 4561 × 6841 × 13681 × 27361 × 54721 a(7) = U(7, 780320) = 4681921 × 9363841 × 14045761 × 28091521 × 56183041 × 112366081 × 224732161 a(8) = U(8, 950560) = 5703361 × 11406721 × 17110081 × 34220161 × 68440321 × 136880641 × 273761281 × 547522561 a(9) = U(9, 950560) = 5703361 × 11406721 × 17110081 × 34220161 × 68440321 × 136880641 × 273761281 × 547522561 × 1095045121 a(10) = U(10, 3208386195840) = 19250317175041 × 38500634350081 × 57750951525121 × 115501903050241 × 231003806100481 × 462007612200961 × 924015224401921 × 1848030448803841 × 3696060897607681 × 7392121795215361
PARI/GP
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))- Output:
cherCar(3): m = 1 cherCar(4): m = 1 cherCar(5): m = 380 cherCar(6): m = 380 cherCar(7): m = 780320 cherCar(8): m = 950560 cherCar(9): m = 950560 cherCar(10): m = 3208386195840
Perl
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) });
}
- Output:
a(3) = 1729 a(4) = 63973 a(5) = 26641259752490421121 a(6) = 1457836374916028334162241 a(7) = 24541683183872873851606952966798288052977151461406721 a(8) = 53487697914261966820654105730041031613370337776541835775672321 a(9) = 58571442634534443082821160508299574798027946748324125518533225605795841
Phix
with javascript_semantics function chernick_carmichael_factors(integer n, m) sequence res = {6*m + 1, 12*m + 1} for i=1 to n-2 do res &= power(2,i) * 9*m + 1 end for return res end function include mpfr.e mpz p = mpz_init() function m_prime(atom a) mpz_set_d(p,a) return mpz_prime(p) end function function is_chernick_carmichael(integer n, m) return iff(n==2 ? m_prime(6*m + 1) and m_prime(12*m + 1) : m_prime(power(2,n-2) * 9*m + 1) and is_chernick_carmichael(n-1, m)) end function function chernick_carmichael_number(integer n) integer m = iff(n>4 ? power(2,n-4) : 1), mm = m while not is_chernick_carmichael(n, mm) do mm += m end while return {chernick_carmichael_factors(n, mm),mm} end function for n=3 to 9 do {sequence f, integer m} = chernick_carmichael_number(n) mpz_set_si(p,1) for i=1 to length(f) do mpz_mul_d(p,p,f[i]) f[i] = sprintf("%d",f[i]) end for printf(1,"U(%d,%d): %s = %s\n",{n,m,mpz_get_str(p),join(f," * ")}) end for
- Output:
U(3,1): 1729 = 7 * 13 * 19 U(4,1): 63973 = 7 * 13 * 19 * 37 U(5,380): 26641259752490421121 = 2281 * 4561 * 6841 * 13681 * 27361 U(6,380): 1457836374916028334162241 = 2281 * 4561 * 6841 * 13681 * 27361 * 54721 U(7,780320): 24541683183872873851606952966798288052977151461406721 = 4681921 * 9363841 * 14045761 * 28091521 * 56183041 * 112366081 * 224732161 U(8,950560): 53487697914261966820654105730041031613370337776541835775672321 = 5703361 * 11406721 * 17110081 * 34220161 * 68440321 * 136880641 * 273761281 * 547522561 U(9,950560): 58571442634534443082821160508299574798027946748324125518533225605795841 = 5703361 * 11406721 * 17110081 * 34220161 * 68440321 * 136880641 * 273761281 * 547522561 * 1095045121
Pleasingly fast, note however that a(10) remains well out of reach / would probably need a complete rewrite.
with cheat
with added cheat for the a(10) case - I found a nice big prime factor of k and added that on each iteration instead of 1.
You could also use the sequence {1,1,1,1,19,19,4877,457,457,12564169}, if you know a way to build that, and then it wouldn't be cheating anymore...
with javascript_semantics include mpfr.e sequence ppp = {3,5,7,11,13,17,19,23} function primality_pretest(atom k) for i=1 to length(ppp) do if remainder(k,ppp[i])=0 then return (k<=23) end if end for return true end function function probprime(atom k, mpz n) mpz_set_d(n, k) return mpz_prime(n) end function function is_chernick(integer n, atom m, mpz z) atom t = 9 * m; if primality_pretest(6 * m + 1) == false then return false end if if primality_pretest(12 * m + 1) == false then return false end if for i=1 to n-3 do if primality_pretest(t*power(2,i) + 1) == false then return false end if end for if probprime(6 * m + 1, z) == false then return false end if if probprime(12 * m + 1, z) == false then return false end if for i=1 to n-2 do if probprime(t*power(2,i) + 1, z) == false then return false end if end for return true end function procedure main() atom t0 = time() mpz z = mpz_init(0) for n=3 to 10 do atom multiplier = iff(n>4 ? power(2,n-4) : 1), k = 1 if n>5 then multiplier *= 5 end if while true do if n=10 then k += 12564168 end if -- cheat! atom m = k * multiplier; if is_chernick(n, m, z) then printf(1,"a(%d) has m = %d\n", {n, m}) exit end if k += 1 end while end for ?elapsed(time()-t0) end procedure main()
- Output:
a(3) has m = 1 a(4) has m = 1 a(5) has m = 380 a(6) has m = 380 a(7) has m = 780320 a(8) has m = 950560 a(9) has m = 950560 a(10) has m = 3208386195840 "0.1s"
Prolog
SWI Prolog is too slow to solve for a(10), even with optimizing by increasing the multiplier and implementing a trial division check. (actually, my implementation of Miller-Rabin in Prolog already starts with a trial division by small primes.)
?- 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.
isprime predicate:
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).
- Output:
3: 7*13*19 = 1729 4: 7*13*19*37 = 63973 5: 2281*4561*6841*13681*27361 = 26641259752490421121 6: 2281*4561*6841*13681*27361*54721 = 1457836374916028334162241 7: 4681921*9363841*14045761*28091521*56183041*112366081*224732161 = 24541683183872873851606952966798288052977151461406721 8: 5703361*11406721*17110081*34220161*68440321*136880641*273761281*547522561 = 53487697914261966820654105730041031613370337776541835775672321 9: 5703361*11406721*17110081*34220161*68440321*136880641*273761281*547522561*1095045121 = 58571442634534443082821160508299574798027946748324125518533225605795841
Python
"""
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
- Output:
a(3) has m = 1 a(4) has m = 1 a(5) has m = 380 a(6) has m = 380 a(7) has m = 780320 a(8) has m = 950560 a(9) has m = 950560
Raku
(formerly Perl 6)
Use the ntheory library from Perl for primality testing since it is much, much faster than Raku's built-in .is-prime method.
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(' ⨉ ')}";
}
- Output:
U(3, 1): 1729 = 7 ⨉ 13 ⨉ 19 U(4, 1): 63973 = 7 ⨉ 13 ⨉ 19 ⨉ 37 U(5, 380): 26641259752490421121 = 2281 ⨉ 4561 ⨉ 6841 ⨉ 13681 ⨉ 27361 U(6, 380): 1457836374916028334162241 = 2281 ⨉ 4561 ⨉ 6841 ⨉ 13681 ⨉ 27361 ⨉ 54721 U(7, 780320): 24541683183872873851606952966798288052977151461406721 = 4681921 ⨉ 9363841 ⨉ 14045761 ⨉ 28091521 ⨉ 56183041 ⨉ 112366081 ⨉ 224732161 U(8, 950560): 53487697914261966820654105730041031613370337776541835775672321 = 5703361 ⨉ 11406721 ⨉ 17110081 ⨉ 34220161 ⨉ 68440321 ⨉ 136880641 ⨉ 273761281 ⨉ 547522561 U(9, 950560): 58571442634534443082821160508299574798027946748324125518533225605795841 = 5703361 ⨉ 11406721 ⨉ 17110081 ⨉ 34220161 ⨉ 68440321 ⨉ 136880641 ⨉ 273761281 ⨉ 547522561 ⨉ 1095045121
REXX
Libraries: How to use
Library: Functions
Library: Numbers
Library: Settings
Library: Abend
Below program uses the improvements as given in Discussion and several other entries. The procedure Prime() is in library Numbers (Miller-Rabin with many tricks).
include Settings
say version; say 'Chernick''s Carmichael numbers'; say
numeric digits 80
say Copies('-',80)
say 'n m(n) a(n)'
say Copies('-',80)
do n = 3 to 9
mp = 1
if n > 4 then
mp = 2**(n-4)
if n > 5 then
mp = mp*5
k = 0
do x = 1
k = k+1; m = mp*k; f.1 = 6*m+1
if \ IsPrime(f.1) then
iterate x
f.2 = 12*m+1
if \ IsPrime(f.2) then
iterate x
f = 2
do i = 1 to n-2
f = f+1; f.f = 2**i*9*m+1
if \ IsPrime(f.f) then
iterate x
end
a = 1
do i = 1 to f
a = a*f.i
end
say n Right(m,6) a
leave x
end
end
say Copies('-',80)
say Format(Time('e'),3,3) 'seconds'
say
exit
include Numbers
include Functions
include Abend
- Output:
REXX-ooRexx_5.0.0(MT)_64-bit 6.05 23 Dec 2022 Chernick's Carmichael numbers -------------------------------------------------------------------------------- n m(n) a(n) -------------------------------------------------------------------------------- 3 1 1729 4 1 63973 5 380 26641259752490421121 6 380 1457836374916028334162241 7 780320 24541683183872873851606952966798288052977151461406721 8 950560 53487697914261966820654105730041031613370337776541835775672321 9 950560 58571442634534443082821160508299574798027946748324125518533225605795841 -------------------------------------------------------------------------------- 1.406 seconds
a(10) was not reachable with REXX. It would run for weeks.
Sidef
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(' * ')}" })
}
- Output:
a(3) = 7 * 13 * 19 a(4) = 7 * 13 * 19 * 37 a(5) = 2281 * 4561 * 6841 * 13681 * 27361 a(6) = 2281 * 4561 * 6841 * 13681 * 27361 * 54721 a(7) = 4681921 * 9363841 * 14045761 * 28091521 * 56183041 * 112366081 * 224732161 a(8) = 5703361 * 11406721 * 17110081 * 34220161 * 68440321 * 136880641 * 273761281 * 547522561 a(9) = 5703361 * 11406721 * 17110081 * 34220161 * 68440321 * 136880641 * 273761281 * 547522561 * 1095045121
Wren
Based on Go's 'more efficient' version. Reaches a(9) in just over 0.1 seconds but a(10) would still be out of reasonable reach for Wren so I've had to be content with that.
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)
- Output:
a(3) = 1729 m(3) = 1 Factors: [7, 13, 19] a(4) = 63973 m(4) = 1 Factors: [7, 13, 19, 37] a(5) = 26641259752490421121 m(5) = 380 Factors: [2281, 4561, 6841, 13681, 27361] a(6) = 1457836374916028334162241 m(6) = 380 Factors: [2281, 4561, 6841, 13681, 27361, 54721] a(7) = 24541683183872873851606952966798288052977151461406721 m(7) = 780320 Factors: [4681921, 9363841, 14045761, 28091521, 56183041, 112366081, 224732161] a(8) = 53487697914261966820654105730041031613370337776541835775672321 m(8) = 950560 Factors: [5703361, 11406721, 17110081, 34220161, 68440321, 136880641, 273761281, 547522561] a(9) = 58571442634534443082821160508299574798027946748324125518533225605795841 m(9) = 950560 Factors: [5703361, 11406721, 17110081, 34220161, 68440321, 136880641, 273761281, 547522561, 1095045121]
zkl
GNU Multiple Precision Arithmetic Library
Using GMP (probabilistic primes), because it is easy and fast to check primeness.
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;
}
}
}ccNumbers(3,9);- Output:
a(3) = 1,729 a(4) = 63,973 a(5) = 26,641,259,752,490,421,121 a(6) = 1,457,836,374,916,028,334,162,241 a(7) = 24,541,683,183,872,873,851,606,952,966,798,288,052,977,151,461,406,721 a(8) = 53,487,697,914,261,966,820,654,105,730,041,031,613,370,337,776,541,835,775,672,321 a(9) = 58,571,442,634,534,443,082,821,160,508,299,574,798,027,946,748,324,125,518,533,225,605,795,841