Chernick's Carmichael numbers

Chernick's Carmichael numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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

• Jack Chernick, On Fermat's simple theorem (PDF)

• OEIS A318646: The least Chernick's "universal form" Carmichael number with n prime factors

Related tasks

• Carmichael 3 strong pseudoprimes

C++

Library: GMP

<lang cpp>#include <gmp.h>

1. 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

Library: GMP(Go wrapper)

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] 

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

Library: ntheory

<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

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 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

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

Translation of: Go

Library: GMP

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