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# Chernick's Carmichael numbers

Chernick's Carmichael numbers
You are encouraged to solve this task according to the task description, using any language you may know.

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.

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
...
```

## C

Library: GMP
`#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++

Library: GMP
`#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#

This task uses Extensible Prime Generator (F#)

` // Generate Chernick's Carmichael numbers. Nigel Galloway: June 1st., 2019let 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 `
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

`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]
```

## 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 trueend 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) == 1end 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 trueend 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     prodend 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    endend 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

Library: bignum

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, sequtilsimport 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

Library: ntheory
`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

Library: Phix/mpfr
Translation of: Sidef
```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

Translation of: C
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, !. % 2^64bases(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 ofhttp://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)

Works with: Rakudo version 2019.03
Translation of: Perl

Use the ntheory library from Perl 5 for primality testing since it is much, much faster than Rakus 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```

## 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

Translation of: Go
Library: Wren-big
Library: Wren-fmt

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, BigIntsimport "/fmt" for Fmt var min = 3var max = 9var prod = BigInt.zerovar fact = BigInt.zerovar 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

Translation of: Go
Library: GMP
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
```