Chernick's Carmichael numbers
In 1939, Jack Chernick proved that, for n ≥ 3 and m ≥ 1:
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
<lang 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;
}</lang>
- 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++
<lang cpp>#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;
}</lang>
- 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#
This task uses Extensible Prime Generator (F#) <lang fsharp> // Generate Chernick's Carmichael numbers. Nigel Galloway: June 1st., 2019 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)) let fX k=Seq.initInfinite(fun n->(n+1)*(pown 2 (k-4))) |> Seq.filter(fun n->fMk n k ) 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)) [4..9] |> Seq.iter cherCar </lang>
- Output:
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]
Go
Basic only
<lang go>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)
}</lang>
- 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. <lang go>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)
}</lang>
- 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]
Java
<lang 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;
}
} </lang>
- 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
<lang 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)</lang>
- 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)
PARI/GP
<lang parigp> 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)) </lang>
- 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
<lang 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($n) = ", vecprod(@f) });
}</lang>
- Output:
a(3) = 1729 a(4) = 63973 a(5) = 26641259752490421121 a(6) = 1457836374916028334162241 a(7) = 24541683183872873851606952966798288052977151461406721 a(8) = 53487697914261966820654105730041031613370337776541835775672321 a(9) = 58571442634534443082821160508299574798027946748324125518533225605795841
Perl 6
Use the ntheory library from Perl 5 for primality testing since it is much, much faster than Perl 6s built-in .is-prime method.
<lang perl6>use Inline::Perl5; use ntheory:from<Perl5> <:all>;
sub chernick-factors ($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(' ⨉ ')}";
}</lang>
- 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
Phix
<lang Phix>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() randstate state = gmp_randinit_mt()
function m_prime(atom a)
mpz_set_d(p,a) return mpz_probable_prime_p(p, state)
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 multiplier = iff(n>4 ? power(2,n-4) : 1), m = 1 while not is_chernick_carmichael(n, m * multiplier) do m += 1 end while return chernick_carmichael_factors(n, m * multiplier)
end function
for n=3 to 9 do
sequence f = chernick_carmichael_number(n)
for i=1 to length(f) do f[i] = sprintf("%d",f[i]) end for
printf(1,"a(%d) = %s\n",{n,join(f," * ")})
end for</lang>
- 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
Pleasingly fast, note however that a(10) remains well out of reach / would probably need a complete rewrite.
Sidef
<lang ruby>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(' * ')}" })
}</lang>
- 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
zkl
GNU Multiple Precision Arithmetic Library
Using GMP (probabilistic primes), because it is easy and fast to check primeness. <lang zkl>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;
} }
}</lang> <lang zkl>ccNumbers(3,9);</lang>
- 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