Carmichael 3 strong pseudoprimes

A lot of composite numbers can be separated from primes by Fermat's Little Theorem, but there are some that completely confound it. The Miller Rabin Test uses a combination of Fermat's Little Theorem and Chinese Division Theorem to overcome this.

The purpose of this task is to investigate such numbers using a method based on Carmichael numbers, as suggested in Notes by G.J.O Jameson March 2010.

The objective is to find Carmichael numbers of the form $Prime_{1}\times Prime_{2}\times Prime_{3}$ (where $Prime_{1}<Prime_{2}<Prime_{3}$ ) for all $Prime_{1}$ up to 61 (see page 7 of Notes by G.J.O Jameson March 2010 for solutions).

Pseudocode:
For a given $Prime_{1}$

for 1 < h3 < Prime₁

for 0 < d < h3+Prime₁

if (h3+Prime₁)*(Prime₁-1) mod d == 0 and -Prime₁ squared mod h3 == d mod h3

then

Prime₂ = 1 + ((Prime₁-1) * (h3+Prime₁)/d)

next d if Prime₂ is not prime

Prime₃ = 1 + (Prime₁*Prime₂/h3)

next d if Prime₃ is not prime

next d if (Prime₂*Prime₃) mod (Prime₁-1) not equal 1

Prime₁ * Prime₂ * Prime₃ is a Carmichael Number

Ada

Uses the Miller_Rabin package from Miller-Rabin primality test#ordinary integers. <lang Ada>with Ada.Text_IO, Miller_Rabin;

procedure Nemesis is

  type Number is range 0 .. 2**40-1; -- sufficiently large for the task

  function Is_Prime(N: Number) return Boolean is
     package MR is new Miller_Rabin(Number); use MR;
  begin
     return MR.Is_Prime(N) = Probably_Prime;
  end Is_Prime;

begin

  for P1 in Number(2) .. 61 loop
     if Is_Prime(P1) then
        for H3 in Number(1) .. P1 loop
           declare
              G: Number := H3 + P1;
              P2, P3: Number;
           begin
              Inner:
              for D in 1 .. G-1 loop
                 if ((H3+P1) * (P1-1)) mod D = 0 and then
                   (-(P1 * P1)) mod H3 = D mod H3
                 then
                    P2 := 1 + ((P1-1) * G / D);
                    P3 := 1 +(P1*P2/H3);
                    if Is_Prime(P2) and then Is_Prime(P3)
                      and then (P2*P3) mod (P1-1) = 1
                    then
                      Ada.Text_IO.Put_Line
                       ( Number'Image(P1) & " *"   & Number'Image(P2) & " *" &
                         Number'Image(P3) & "  = " & Number'Image(P1*P2*P3) );
                    end if;
                 end if;
              end loop Inner;
           end;
        end loop;
     end if;
  end loop;

end Nemesis;</lang>

Output:

 3 * 11 * 17  =  561
 5 * 29 * 73  =  10585
 5 * 17 * 29  =  2465
 5 * 13 * 17  =  1105
 7 * 19 * 67  =  8911

... (the full output is 69 lines long) ...

 61 * 271 * 571  =  9439201
 61 * 241 * 421  =  6189121
 61 * 3361 * 4021  =  824389441

C

include <stdio.h>

/* C's % operator actually calculates the remainder of a / b so we need a

* small adjustment so it works as expected for negative values */

define mod(n,m) ((((n) % (m)) + (m)) % (m))

int is_prime(unsigned int n) {

   if (n <= 3) {
       return n > 1;
   }
   else if (!(n % 2) || !(n % 3)) {
       return 0;
   }
   else {
       unsigned int i;
       for (i = 5; i*i <= n; i += 6)
           if (!(n % i) || !(n % (i + 2)))
               return 0;
       return 1;
   }

}

void carmichael3(int p1) {

   if (!is_prime(p1)) return;

   int h3, d, p2, p3;
   for (h3 = 1; h3 < p1; ++h3) {
       for (d = 1; d < h3 + p1; ++d) {
           if ((h3 + p1)*(p1 - 1) % d == 0 && mod(-p1 * p1, h3) == d % h3) {
               p2 = 1 + ((p1 - 1) * (h3 + p1)/d);
               if (!is_prime(p2)) continue;
               p3 = 1 + (p1 * p2 / h3);
               if (!is_prime(p3) || (p2 * p3) % (p1 - 1) != 1) continue;
               printf("%d %d %d\n", p1, p2, p3);
           }
       }
   }

}

int main(void) {

   int p1;
   for (p1 = 2; p1 < 62; ++p1)
       carmichael3(p1);
   return 0;

} </lang>

Output:

3 11 17
5 29 73
5 17 29
5 13 17
7 19 67
7 31 73
.
.
.
61 181 1381
61 541 3001
61 661 2521
61 271 571
61 241 421
61 3361 4021

D

<lang d>enum mod = (in int n, in int m) pure nothrow @nogc=> ((n % m) + m) % m;

bool isPrime(in uint n) pure nothrow @nogc {

 if (n == 2 || n == 3)
   return true;
 else if (n < 2 || n % 2 == 0 || n % 3 == 0)
   return false;
 for (uint div = 5, inc = 2; div ^^ 2 <= n;
    div += inc, inc = 6 - inc)
   if (n % div == 0)
     return false;
 return true;

}

void main() {

 import std.stdio;

 foreach (immutable p; 2 .. 62) {
   if (!p.isPrime) continue;
   foreach (immutable h3; 2 .. p) {
     immutable g = h3 + p;
     foreach (immutable d; 1 .. g) {
       if ((g * (p - 1)) % d != 0 || mod(-p * p, h3) != d % h3)
         continue;
       immutable q = 1 + (p - 1) * g / d;
       if (!q.isPrime) continue;
       immutable r = 1 + (p * q / h3);
       if (!r.isPrime || (q * r) % (p - 1) != 1) continue;
       writeln(p, " x ", q, " x ", r);
     }
   }
 }

}</lang>

Output:

3 x 11 x 17
5 x 29 x 73
5 x 17 x 29
5 x 13 x 17
7 x 19 x 67
7 x 31 x 73
7 x 13 x 31
7 x 23 x 41
7 x 73 x 103
7 x 13 x 19
13 x 61 x 397
13 x 37 x 241
13 x 97 x 421
13 x 37 x 97
13 x 37 x 61
17 x 41 x 233
17 x 353 x 1201
19 x 43 x 409
19 x 199 x 271
23 x 199 x 353
29 x 113 x 1093
29 x 197 x 953
31 x 991 x 15361
31 x 61 x 631
31 x 151 x 1171
31 x 61 x 271
31 x 61 x 211
31 x 271 x 601
31 x 181 x 331
37 x 109 x 2017
37 x 73 x 541
37 x 613 x 1621
37 x 73 x 181
37 x 73 x 109
41 x 1721 x 35281
41 x 881 x 12041
41 x 101 x 461
41 x 241 x 761
41 x 241 x 521
41 x 73 x 137
41 x 61 x 101
43 x 631 x 13567
43 x 271 x 5827
43 x 127 x 2731
43 x 127 x 1093
43 x 211 x 757
43 x 631 x 1597
43 x 127 x 211
43 x 211 x 337
43 x 433 x 643
43 x 547 x 673
43 x 3361 x 3907
47 x 3359 x 6073
47 x 1151 x 1933
47 x 3727 x 5153
53 x 157 x 2081
53 x 79 x 599
53 x 157 x 521
59 x 1451 x 2089
61 x 421 x 12841
61 x 181 x 5521
61 x 1301 x 19841
61 x 277 x 2113
61 x 181 x 1381
61 x 541 x 3001
61 x 661 x 2521
61 x 271 x 571
61 x 241 x 421
61 x 3361 x 4021

Haskell

Translation of: Ruby

Library: primes

Works with: GHC version 7.4.1

Works with: primes version 0.2.1.0

<lang haskell>#!/usr/bin/runhaskell

import Data.Numbers.Primes import Control.Monad (guard)

carmichaels = do

 p <- takeWhile (<= 61) primes
 h3 <- [2..(p-1)]
 let g = h3 + p
 d <- [1..(g-1)]
 guard $ (g * (p - 1)) `mod` d == 0 && (-1 * p * p) `mod` h3 == d `mod` h3
 let q = 1 + (((p - 1) * g) `div` d)
 guard $ isPrime q
 let r = 1 + ((p * q) `div` h3)
 guard $ isPrime r && (q * r) `mod` (p - 1) == 1
 return (p, q, r)

main = putStr $ unlines $ map show carmichaels</lang>

Output:

(3,11,17)
(5,29,73)
(5,17,29)
(5,13,17)
(7,19,67)
(7,31,73)
(7,13,31)
(7,23,41)
(7,73,103)
(7,13,19)
(13,61,397)
(13,37,241)
(13,97,421)
(13,37,97)
(13,37,61)
(17,41,233)
(17,353,1201)
(19,43,409)
(19,199,271)
(23,199,353)
(29,113,1093)
(29,197,953)
(31,991,15361)
(31,61,631)
(31,151,1171)
(31,61,271)
(31,61,211)
(31,271,601)
(31,181,331)
(37,109,2017)
(37,73,541)
(37,613,1621)
(37,73,181)
(37,73,109)
(41,1721,35281)
(41,881,12041)
(41,101,461)
(41,241,761)
(41,241,521)
(41,73,137)
(41,61,101)
(43,631,13567)
(43,271,5827)
(43,127,2731)
(43,127,1093)
(43,211,757)
(43,631,1597)
(43,127,211)
(43,211,337)
(43,433,643)
(43,547,673)
(43,3361,3907)
(47,3359,6073)
(47,1151,1933)
(47,3727,5153)
(53,157,2081)
(53,79,599)
(53,157,521)
(59,1451,2089)
(61,421,12841)
(61,181,5521)
(61,1301,19841)
(61,277,2113)
(61,181,1381)
(61,541,3001)
(61,661,2521)
(61,271,571)
(61,241,421)
(61,3361,4021)

Icon and Unicon

The following works in both languages. <lang unicon>link "factors"

procedure main(A)

   n := integer(!A) | 61
   every write(carmichael3(!n))

end

procedure carmichael3(p1)

   every (isprime(p1), (h := 1+!(p1-1)), (d := !(h+p1-1))) do
       if (mod(((h+p1)*(p1-1)),d) = 0, mod((-p1*p1),h) = mod(d,h)) then {
           p2 := 1 + (p1-1)*(h+p1)/d
           p3 := 1 + p1*p2/h
           if (isprime(p2), isprime(p3), mod((p2*p3),(p1-1)) = 1) then
               suspend format(p1,p2,p3)
           }

end

procedure mod(n,d)

  return (d+n%d)%d

end

procedure format(p1,p2,p3)

   return left(p1||" * "||p2||" * "||p3,20)||" = "||(p1*p2*p3)

end</lang>

Output, with middle lines elided:

->c3sp
3 * 11 * 17          = 561
5 * 29 * 73          = 10585
5 * 17 * 29          = 2465
5 * 13 * 17          = 1105
7 * 19 * 67          = 8911
7 * 31 * 73          = 15841
7 * 13 * 31          = 2821
7 * 23 * 41          = 6601
7 * 73 * 103         = 52633
7 * 13 * 19          = 1729
13 * 61 * 397        = 314821
13 * 37 * 241        = 115921
...
53 * 157 * 2081      = 17316001
53 * 79 * 599        = 2508013
53 * 157 * 521       = 4335241
59 * 1451 * 2089     = 178837201
61 * 421 * 12841     = 329769721
61 * 181 * 5521      = 60957361
61 * 1301 * 19841    = 1574601601
61 * 277 * 2113      = 35703361
61 * 181 * 1381      = 15247621
61 * 541 * 3001      = 99036001
61 * 661 * 2521      = 101649241
61 * 271 * 571       = 9439201
61 * 241 * 421       = 6189121
61 * 3361 * 4021     = 824389441
->

Julia

This solution is a straightforward implementation of the algorithm of the Jameson paper cited in the task description. Just for fun, I use Julia's capacity to accommodate Unicode identifiers to match some of the paper's symbols to the variables used in the carmichael function.

Function <lang Julia> function carmichael{T<:Integer}(pmax::T)

   0 < pmax || throw(DomainError())
   car = T[]
   for p in primes(pmax)
       for h₃ in 2:(p-1)
           m = (p - 1)*(h₃ + p)
           pmh = mod(-p^2, h₃)
           for Δ in 1:(h₃+p-1)
               m%Δ==0 && Δ%h₃==pmh || continue
               q = div(m, Δ) + 1
               isprime(q) || continue
               r = div((p*q - 1), h₃) + 1
               isprime(r) && mod(q*r, (p-1))==1 || continue
               append!(car, [p, q, r])
           end
       end
   end
   reshape(car, 3, div(length(car), 3))

end </lang>

Main <lang Julia> hi = 61 car = carmichael(hi)

curp = 0 tcnt = 0 print("Carmichael 3 (p\u00d7q\u00d7r) Pseudoprimes, up to p = ", hi, ":") for j in sortperm(1:size(car)[2], by=x->(car[1,x], car[2,x], car[3,x]))

   p, q, r = car[:,j]
   c = prod(car[:,j])
   if p != curp
       curp = p
       print(@sprintf("\n\np = %d\n  ", p))
       tcnt = 0
   end
   if tcnt == 4
       print("\n  ")
       tcnt = 1
   else
       tcnt += 1
   end
   print(@sprintf("p\u00d7%d\u00d7%d = %d  ", q, r, c))

end println("\n\n", size(car)[2], " results in total.") </lang>

Output:

Carmichael 3 (p×q×r) Pseudoprimes, up to p = 61:

p = 3
  p×11×17 = 561  

p = 5
  p×13×17 = 1105  p×17×29 = 2465  p×29×73 = 10585  

p = 7
  p×13×19 = 1729  p×13×31 = 2821  p×19×67 = 8911  p×23×41 = 6601  
  p×31×73 = 15841  p×73×103 = 52633  

p = 13
  p×37×61 = 29341  p×37×97 = 46657  p×37×241 = 115921  p×61×397 = 314821  
  p×97×421 = 530881  

p = 17
  p×41×233 = 162401  p×353×1201 = 7207201  

p = 19
  p×43×409 = 334153  p×199×271 = 1024651  

p = 23
  p×199×353 = 1615681  

p = 29
  p×113×1093 = 3581761  p×197×953 = 5444489  

p = 31
  p×61×211 = 399001  p×61×271 = 512461  p×61×631 = 1193221  p×151×1171 = 5481451  
  p×181×331 = 1857241  p×271×601 = 5049001  p×991×15361 = 471905281  

p = 37
  p×73×109 = 294409  p×73×181 = 488881  p×73×541 = 1461241  p×109×2017 = 8134561  
  p×613×1621 = 36765901  

p = 41
  p×61×101 = 252601  p×73×137 = 410041  p×101×461 = 1909001  p×241×521 = 5148001  
  p×241×761 = 7519441  p×881×12041 = 434932961  p×1721×35281 = 2489462641  

p = 43
  p×127×211 = 1152271  p×127×1093 = 5968873  p×127×2731 = 14913991  p×211×337 = 3057601  
  p×211×757 = 6868261  p×271×5827 = 67902031  p×433×643 = 11972017  p×547×673 = 15829633  
  p×631×1597 = 43331401  p×631×13567 = 368113411  p×3361×3907 = 564651361  

p = 47
  p×1151×1933 = 104569501  p×3359×6073 = 958762729  p×3727×5153 = 902645857  

p = 53
  p×79×599 = 2508013  p×157×521 = 4335241  p×157×2081 = 17316001  

p = 59
  p×1451×2089 = 178837201  

p = 61
  p×181×1381 = 15247621  p×181×5521 = 60957361  p×241×421 = 6189121  p×271×571 = 9439201  
  p×277×2113 = 35703361  p×421×12841 = 329769721  p×541×3001 = 99036001  p×661×2521 = 101649241  
  p×1301×19841 = 1574601601  p×3361×4021 = 824389441  

69 results in total.

Mathematica

<lang mathematica>Cases[Cases[

 Cases[Table[{p1, h3, d}, {p1, Array[Prime, PrimePi@61]}, {h3, 2, 
    p1 - 1}, {d, 1, h3 + p1 - 1}], {p1_Integer, h3_, d_} /; 
    PrimeQ[1 + (p1 - 1) (h3 + p1)/d] && 
     Divisible[p1^2 + d, h3] :> {p1, 1 + (p1 - 1) (h3 + p1)/d, h3}, 
  Infinity], {p1_, p2_, h3_} /; PrimeQ[1 + Floor[p1 p2/h3]] :> {p1, 
   p2, 1 + Floor[p1 p2/h3]}], {p1_, p2_, p3_} /; 
  Mod[p2 p3, p1 - 1] == 1 :> Print[p1, "*", p2, "*", p3]]</lang>

PARI/GP

<lang parigp>f(p)={

 my(v=List(),q,r);
 for(h=2,p-1,
   for(d=1,h+p-1,
     if((h+p)*(p-1)%d==0 && Mod(p,h)^2==-d && isprime(q=(p-1)*(h+p)/d+1) && isprime(r=p*q\h+1)&&q*r%(p-1)==1,
       listput(v,p*q*r)
     )
   )
 );
 Set(v)

}; forprime(p=3,67,v=f(p); for(i=1,#v,print1(v[i]", ")))</lang>

Output:

561, 1105, 2465, 10585, 1729, 2821, 6601, 8911, 15841, 52633, 29341, 46657, 115921, 314821, 530881, 162401, 7207201, 334153, 1024651, 1615681, 3581761, 5444489, 399001, 512461, 1193221, 1857241, 5049001, 5481451, 471905281, 294409, 488881, 1461241, 8134561, 36765901, 252601, 410041, 1909001, 5148001, 7519441, 434932961, 2489462641, 1152271, 3057601, 5968873, 6868261, 11972017, 14913991, 15829633, 43331401, 67902031, 368113411, 564651361, 104569501, 902645857, 958762729, 2508013, 4335241, 17316001, 178837201, 6189121, 9439201, 15247621, 35703361, 60957361, 99036001, 101649241, 329769721, 824389441, 1574601601, 10267951, 163954561, 7991602081,

Perl

Library: ntheory

<lang perl>use ntheory qw/forprimes is_prime vecprod/;

forprimes { my $p = $_;

  for my $h3 (2 .. $p-1) {
     my $ph3 = $p + $h3;
     for my $d (1 .. $ph3-1) {               # Jameseon procedure page 6
        next if ((-$p*$p) % $h3) != ($d % $h3);
        next if (($p-1)*$ph3) % $d;
        my $q = 1 + ($p-1)*$ph3 / $d;        # Jameson eq 7
        next unless is_prime($q);
        my $r = 1 + ($p*$q-1) / $h3;         # Jameson eq 6
        next unless is_prime($r);
        next unless ($q*$r) % ($p-1) == 1;
        printf "%2d x %5d x %5d = %s\n",$p,$q,$r,vecprod($p,$q,$r);
     }
  }

} 3,61;</lang>

Output:

 3 x    11 x    17 = 561
 5 x    29 x    73 = 10585
 5 x    17 x    29 = 2465
 5 x    13 x    17 = 1105
 ... full output is 69 lines ...
61 x   661 x  2521 = 101649241
61 x   271 x   571 = 9439201
61 x   241 x   421 = 6189121
61 x  3361 x  4021 = 824389441

Perl 6

An almost direct translation of the pseudocode. We take the liberty of going up to 67 to show we aren't limited to 32-bit integers. (Perl 6 uses arbitrary precision in any case.) <lang perl6>for (2..67).grep: *.is-prime -> \Prime1 {

   for 1 ^..^ Prime1 -> \h3 {
       my \g = h3 + Prime1;
       for 0 ^..^ h3 + Prime1 -> \d {
           if (h3 + Prime1) * (Prime1 - 1) %% d and -Prime1**2 % h3 == d % h3  {
               my \Prime2 = floor 1 + (Prime1 - 1) * g / d;
               next unless Prime2.is-prime;
               my \Prime3 = floor 1 + Prime1 * Prime2 / h3;
               next unless Prime3.is-prime;
               next unless (Prime2 * Prime3) % (Prime1 - 1) == 1;
               say "{Prime1} × {Prime2} × {Prime3} == {Prime1 * Prime2 * Prime3}";
           }
       }
   }

}</lang>

Output:

3 × 11 × 17 == 561
5 × 29 × 73 == 10585
5 × 17 × 29 == 2465
5 × 13 × 17 == 1105
7 × 19 × 67 == 8911
7 × 31 × 73 == 15841
7 × 13 × 31 == 2821
7 × 23 × 41 == 6601
7 × 73 × 103 == 52633
7 × 13 × 19 == 1729
13 × 61 × 397 == 314821
13 × 37 × 241 == 115921
13 × 97 × 421 == 530881
13 × 37 × 97 == 46657
13 × 37 × 61 == 29341
17 × 41 × 233 == 162401
17 × 353 × 1201 == 7207201
19 × 43 × 409 == 334153
19 × 199 × 271 == 1024651
23 × 199 × 353 == 1615681
29 × 113 × 1093 == 3581761
29 × 197 × 953 == 5444489
31 × 991 × 15361 == 471905281
31 × 61 × 631 == 1193221
31 × 151 × 1171 == 5481451
31 × 61 × 271 == 512461
31 × 61 × 211 == 399001
31 × 271 × 601 == 5049001
31 × 181 × 331 == 1857241
37 × 109 × 2017 == 8134561
37 × 73 × 541 == 1461241
37 × 613 × 1621 == 36765901
37 × 73 × 181 == 488881
37 × 73 × 109 == 294409
41 × 1721 × 35281 == 2489462641
41 × 881 × 12041 == 434932961
41 × 101 × 461 == 1909001
41 × 241 × 761 == 7519441
41 × 241 × 521 == 5148001
41 × 73 × 137 == 410041
41 × 61 × 101 == 252601
43 × 631 × 13567 == 368113411
43 × 271 × 5827 == 67902031
43 × 127 × 2731 == 14913991
43 × 127 × 1093 == 5968873
43 × 211 × 757 == 6868261
43 × 631 × 1597 == 43331401
43 × 127 × 211 == 1152271
43 × 211 × 337 == 3057601
43 × 433 × 643 == 11972017
43 × 547 × 673 == 15829633
43 × 3361 × 3907 == 564651361
47 × 3359 × 6073 == 958762729
47 × 1151 × 1933 == 104569501
47 × 3727 × 5153 == 902645857
53 × 157 × 2081 == 17316001
53 × 79 × 599 == 2508013
53 × 157 × 521 == 4335241
59 × 1451 × 2089 == 178837201
61 × 421 × 12841 == 329769721
61 × 181 × 5521 == 60957361
61 × 1301 × 19841 == 1574601601
61 × 277 × 2113 == 35703361
61 × 181 × 1381 == 15247621
61 × 541 × 3001 == 99036001
61 × 661 × 2521 == 101649241
61 × 271 × 571 == 9439201
61 × 241 × 421 == 6189121
61 × 3361 × 4021 == 824389441
67 × 2311 × 51613 == 7991602081
67 × 331 × 7393 == 163954561
67 × 331 × 463 == 10267951

PicoLisp

<lang PicoLisp>(de modulo (X Y)

  (% (+ Y (% X Y)) Y) )

(de prime? (N)

  (let D 0
     (or
        (= N 2)
        (and
           (> N 1)
           (bit? 1 N)
           (for (D 3  T  (+ D 2))
              (T (> D (sqrt N)) T)
              (T (=0 (% N D)) NIL) ) ) ) ) )

(for P1 61

  (when (prime? P1)
     (for (H3 2 (> P1 H3) (inc H3))
        (let G (+ H3 P1)
           (for (D 1 (> G D) (inc D))
              (when
                 (and
                    (=0
                       (% (* G (dec P1)) D) )
                    (=
                       (modulo (* (- P1) P1) H3)
                       (% D H3)) )
                 (let
                    (P2
                       (inc
                          (/ (* (dec P1) G) D) )
                       P3 (inc (/ (* P1 P2) H3)) )
                    (when
                       (and
                          (prime? P2)
                          (prime? P3)
                          (= 1 (modulo (* P2 P3) (dec P1))) )
                       (print (list P1 P2 P3)) ) ) ) ) ) ) ) )

(prinl)

(bye)</lang>

PL/I

<lang PL/I>Carmichael: procedure options (main, reorder); /* 24 January 2014 */

  declare (Prime1, Prime2, Prime3, h3, d) fixed binary (31);

  put ('Carmichael numbers are:');

  do Prime1 = 1 to 61;

     do h3 = 2 to Prime1;

d_loop: do d = 1 to h3+Prime1-1;

           if (mod((h3+Prime1)*(Prime1-1), d) = 0) &
              (mod(-Prime1*Prime1, h3) = mod(d, h3)) then
              do;
                 Prime2 = (Prime1-1) * (h3+Prime1)/d; Prime2 = Prime2 + 1;
                 if ^is_prime(Prime2) then iterate d_loop;
                 Prime3 = Prime1*Prime2/h3; Prime3 = Prime3 + 1;
                 if ^is_prime(Prime3) then iterate d_loop;
                 if mod(Prime2*Prime3, Prime1-1) ^= 1 then iterate d_loop;
                 put skip edit (trim(Prime1), ' x ', trim(Prime2), ' x ', trim(Prime3)) (A);
              end;
        end;
     end;
  end;

  /* Uses is_prime from Rosetta Code PL/I. */

end Carmichael;</lang> Results:

Carmichael numbers are: 
3 x 11 x 17
5 x 29 x 73
5 x 17 x 29
5 x 13 x 17
7 x 19 x 67
7 x 31 x 73
7 x 13 x 31
7 x 23 x 41
7 x 73 x 103
7 x 13 x 19
9 x 89 x 401
9 x 29 x 53
13 x 61 x 397
13 x 37 x 241
13 x 97 x 421
13 x 37 x 97
13 x 37 x 61
17 x 41 x 233
17 x 353 x 1201
19 x 43 x 409
19 x 199 x 271
21 x 761 x 941
23 x 199 x 353
27 x 131 x 443
27 x 53 x 131
29 x 113 x 1093
29 x 197 x 953
31 x 991 x 15361
31 x 61 x 631
31 x 151 x 1171
31 x 61 x 271
31 x 61 x 211
31 x 271 x 601
31 x 181 x 331
35 x 647 x 7549
35 x 443 x 3877
37 x 109 x 2017
37 x 73 x 541
37 x 613 x 1621
37 x 73 x 181
37 x 73 x 109
41 x 1721 x 35281
41 x 881 x 12041
41 x 101 x 461
41 x 241 x 761
41 x 241 x 521
41 x 73 x 137
41 x 61 x 101
43 x 631 x 13567
43 x 271 x 5827
43 x 127 x 2731
43 x 127 x 1093
43 x 211 x 757
43 x 631 x 1597
43 x 127 x 211
43 x 211 x 337
43 x 433 x 643
43 x 547 x 673
43 x 3361 x 3907
47 x 3359 x 6073
47 x 1151 x 1933
47 x 3727 x 5153
49 x 313 x 5113
49 x 97 x 433
51 x 701 x 7151
53 x 157 x 2081
53 x 79 x 599
53 x 157 x 521
55 x 3079 x 84673
55 x 163 x 4483
55 x 1567 x 28729
55 x 109 x 1999
55 x 433 x 2647
55 x 919 x 3889
55 x 139 x 547
55 x 3889 x 12583
55 x 109 x 163
55 x 433 x 487
57 x 113 x 1289
57 x 113 x 281
57 x 4649 x 10193
59 x 1451 x 2089
61 x 421 x 12841
61 x 181 x 5521
61 x 1301 x 19841
61 x 277 x 2113
61 x 181 x 1381
61 x 541 x 3001
61 x 661 x 2521
61 x 271 x 571
61 x 241 x 421
61 x 3361 x 4021

Python

<lang python>class Isprime():

   
   Extensible sieve of Eratosthenes
   
   >>> isprime.check(11)
   True
   >>> isprime.multiples
   {2, 4, 6, 8, 9, 10}
   >>> isprime.primes
   [2, 3, 5, 7, 11]
   >>> isprime(13)
   True
   >>> isprime.multiples
   {2, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22}
   >>> isprime.primes
   [2, 3, 5, 7, 11, 13, 17, 19]
   >>> isprime.nmax
   22
   >>> 
   
   multiples = {2}
   primes = [2]
   nmax = 2
   
   def __init__(self, nmax):
       if nmax > self.nmax:
           self.check(nmax)

   def check(self, n):
       if type(n) == float:
           if not n.is_integer(): return False
           n = int(n)
       multiples = self.multiples
       if n <= self.nmax:
           return n not in multiples
       else:
           # Extend the sieve
           primes, nmax = self.primes, self.nmax
           newmax = max(nmax*2, n)
           for p in primes:
               multiples.update(range(p*((nmax + p + 1) // p), newmax+1, p))
           for i in range(nmax+1, newmax+1):
               if i not in multiples:
                   primes.append(i)
                   multiples.update(range(i*2, newmax+1, i))
           self.nmax = newmax
           return n not in multiples

   __call__ = check

def carmichael(p1):

   ans = []
   if isprime(p1):
       for h3 in range(2, p1):
           g = h3 + p1
           for d in range(1, g):
               if (g * (p1 - 1)) % d == 0 and (-p1 * p1) % h3 == d % h3:
                   p2 = 1 + ((p1 - 1)* g // d)
                   if isprime(p2):
                       p3 = 1 + (p1 * p2 // h3)
                       if isprime(p3):
                           if (p2 * p3) % (p1 - 1) == 1:
                               #print('%i X %i X %i' % (p1, p2, p3))
                               ans += [tuple(sorted((p1, p2, p3)))]
   return ans

isprime = Isprime(2)

ans = sorted(sum((carmichael(n) for n in range(62) if isprime(n)), [])) print(',\n'.join(repr(ans[i:i+5])[1:-1] for i in range(0, len(ans)+1, 5)))</lang>

Output:

(3, 11, 17), (5, 13, 17), (5, 17, 29), (5, 29, 73), (7, 13, 19),
(7, 13, 31), (7, 19, 67), (7, 23, 41), (7, 31, 73), (7, 73, 103),
(13, 37, 61), (13, 37, 97), (13, 37, 241), (13, 61, 397), (13, 97, 421),
(17, 41, 233), (17, 353, 1201), (19, 43, 409), (19, 199, 271), (23, 199, 353),
(29, 113, 1093), (29, 197, 953), (31, 61, 211), (31, 61, 271), (31, 61, 631),
(31, 151, 1171), (31, 181, 331), (31, 271, 601), (31, 991, 15361), (37, 73, 109),
(37, 73, 181), (37, 73, 541), (37, 109, 2017), (37, 613, 1621), (41, 61, 101),
(41, 73, 137), (41, 101, 461), (41, 241, 521), (41, 241, 761), (41, 881, 12041),
(41, 1721, 35281), (43, 127, 211), (43, 127, 1093), (43, 127, 2731), (43, 211, 337),
(43, 211, 757), (43, 271, 5827), (43, 433, 643), (43, 547, 673), (43, 631, 1597),
(43, 631, 13567), (43, 3361, 3907), (47, 1151, 1933), (47, 3359, 6073), (47, 3727, 5153),
(53, 79, 599), (53, 157, 521), (53, 157, 2081), (59, 1451, 2089), (61, 181, 1381),
(61, 181, 5521), (61, 241, 421), (61, 271, 571), (61, 277, 2113), (61, 421, 12841),
(61, 541, 3001), (61, 661, 2521), (61, 1301, 19841), (61, 3361, 4021)

Racket

lang racket

(require math)

(for ([p1 (in-range 3 62)] #:when (prime? p1))

 (for ([h3 (in-range 2 p1)])
   (define g (+ p1 h3))
   (let next ([d 1])
     (when (< d g)
       (when (and (zero? (modulo (* g (- p1 1)) d))
                  (= (modulo (- (sqr p1)) h3) (modulo d h3)))
         (define p2 (+ 1 (quotient (* g (- p1 1)) d)))
         (when (prime? p2)
           (define p3 (+ 1 (quotient (* p1 p2) h3)))
           (when (and (prime? p3) (= 1 (modulo (* p2 p3) (- p1 1))))
             (displayln (list p1 p2 p3 '=> (* p1 p2 p3))))))
       (next (+ d 1))))))

</lang> Output: <lang racket> (3 11 17 => 561) (5 29 73 => 10585) (5 17 29 => 2465) (5 13 17 => 1105) (7 19 67 => 8911) (7 31 73 => 15841) (7 23 41 => 6601) (7 73 103 => 52633) (13 61 397 => 314821) (13 97 421 => 530881) (13 37 97 => 46657) (13 37 61 => 29341) (17 41 233 => 162401) (17 353 1201 => 7207201) (19 43 409 => 334153) (19 199 271 => 1024651) (23 199 353 => 1615681) (29 113 1093 => 3581761) (29 197 953 => 5444489) (31 991 15361 => 471905281) (31 61 631 => 1193221) (31 151 1171 => 5481451) (31 61 271 => 512461) (31 61 211 => 399001) (31 271 601 => 5049001) (31 181 331 => 1857241) (37 109 2017 => 8134561) (37 73 541 => 1461241) (37 613 1621 => 36765901) (37 73 181 => 488881) (37 73 109 => 294409) (41 1721 35281 => 2489462641) (41 881 12041 => 434932961) (41 101 461 => 1909001) (41 241 761 => 7519441) (41 241 521 => 5148001) (41 73 137 => 410041) (41 61 101 => 252601) (43 631 13567 => 368113411) (43 127 1093 => 5968873) (43 211 757 => 6868261) (43 631 1597 => 43331401) (43 127 211 => 1152271) (43 211 337 => 3057601) (43 433 643 => 11972017) (43 547 673 => 15829633) (43 3361 3907 => 564651361) (47 3359 6073 => 958762729) (47 1151 1933 => 104569501) (47 3727 5153 => 902645857) (53 157 2081 => 17316001) (53 79 599 => 2508013) (53 157 521 => 4335241) (59 1451 2089 => 178837201) (61 421 12841 => 329769721) (61 1301 19841 => 1574601601) (61 277 2113 => 35703361) (61 541 3001 => 99036001) (61 661 2521 => 101649241) (61 271 571 => 9439201) (61 241 421 => 6189121) (61 3361 4021 => 824389441) </lang>

REXX

Note that REXX's version of modulus (//) is really a remainder function,
so a version of the modulus function was hard-coded below.
(It was necessary to use modulus instead of remainder when using a negative value.)

The Carmichael numbers are shown in numerical order.
Some code optimization was done, while not necessary for the small default number (61), it was significant for larger numbers. <lang rexx>/*REXX program calculates Carmichael 3-strong pseudoprimes (up to N).*/ numeric digits 30 /*in case user wants bigger nums.*/ parse arg N .; if N== then N=61 /*allow user to specify the limit*/ if 7=='f7'x then times='af'x /*if EBCDIC machine, use a bullet*/

            else times='f9'x          /* " ASCII     "      "  "   "   */

carms=0 /*number of Carmichael #s so far.*/ !.=0; !.2=1; !.3=1; !.5=1; !.7=1; !.11=1; !.13=1; !.17=1; !.19=1; !.23=1

                                      /*[↓]  prime # memoization array.*/
   do p=3  to N  by 2;  if \isPrime(p)  then iterate  /*Not prime? Skip*/
   pm=p-1;  nps=-p*p;   bot=0;  top=0 /*some handy-dandy REXX variables*/
   @.=0                               /*[↑]  Carmichael numbers are odd*/
            do h3=2  to  pm;  g=h3+p  /*find Carmichael #s for this P. */
            gPM=g*pm;         npsH3=((nps//h3)+h3)//h3     /*shortcuts.*/

               do d=1  for g-1
               if gPM//d  \==  0              then iterate
               if npsH3   \==  d//h3          then iterate
               q=1+gPM%d;     if \isPrime(q)  then iterate
               r=1+p*q%h3;    if q*r//pm\==1  then iterate
                              if \isPrime(r)  then iterate
               carms=carms+1; @.q=r   /*bump Carmichael #; add to array*/
               if bot==0  then bot=q;   bot=min(bot,q);    top=max(top,q)
  /*find the maximum.              */
               end   /*d*/            /* [↑]  find minimum and maximum.*/
            end      /*h3*/
   $=0                                /*display a list of some Carm #s.*/
        do j=bot  to top  by 2;       if @.j==0  then iterate;        $=1
        say '──────── a Carmichael number: '   p   times  j   times   @.j
        end   /*j*/
   if $  then say                     /*show beautification blank line.*/
   end        /*p*/

say; say carms ' Carmichael numbers found.' exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────ISPRIME subroutine──────────────────*/ isPrime: procedure expose !.; parse arg x; if !.x then return 1 if x<23 then return 0; if x//2==0 then return 0; if x// 3==0 then return 0 if right(x,1)==5 then return 0; if x// 7==0 then return 0 if x//11==0 then return 0; if x//13==0 then return 0 if x//17==0 then return 0; if x//19==0 then return 0

                  do i=23 by 6  until i*i>x; if x// i   ==0 then return 0
                                             if x//(i+2)==0 then return 0
                  end  /*i*/

!.x=1; return 1</lang> output when using the default input:

──────── a Carmichael number:  3 ∙ 11 ∙ 17

──────── a Carmichael number:  5 ∙ 13 ∙ 17
──────── a Carmichael number:  5 ∙ 17 ∙ 29
──────── a Carmichael number:  5 ∙ 29 ∙ 73

──────── a Carmichael number:  7 ∙ 13 ∙ 19
──────── a Carmichael number:  7 ∙ 19 ∙ 67
──────── a Carmichael number:  7 ∙ 23 ∙ 41
──────── a Carmichael number:  7 ∙ 31 ∙ 73
──────── a Carmichael number:  7 ∙ 73 ∙ 103

──────── a Carmichael number:  13 ∙ 37 ∙ 61
──────── a Carmichael number:  13 ∙ 61 ∙ 397
──────── a Carmichael number:  13 ∙ 97 ∙ 421

──────── a Carmichael number:  17 ∙ 41 ∙ 233
──────── a Carmichael number:  17 ∙ 353 ∙ 1201

──────── a Carmichael number:  19 ∙ 43 ∙ 409
──────── a Carmichael number:  19 ∙ 199 ∙ 271

──────── a Carmichael number:  23 ∙ 199 ∙ 353

──────── a Carmichael number:  29 ∙ 113 ∙ 1093
──────── a Carmichael number:  29 ∙ 197 ∙ 953

──────── a Carmichael number:  31 ∙ 61 ∙ 211
──────── a Carmichael number:  31 ∙ 151 ∙ 1171
──────── a Carmichael number:  31 ∙ 181 ∙ 331
──────── a Carmichael number:  31 ∙ 271 ∙ 601
──────── a Carmichael number:  31 ∙ 991 ∙ 15361

──────── a Carmichael number:  37 ∙ 73 ∙ 109
──────── a Carmichael number:  37 ∙ 109 ∙ 2017
──────── a Carmichael number:  37 ∙ 613 ∙ 1621

──────── a Carmichael number:  41 ∙ 61 ∙ 101
──────── a Carmichael number:  41 ∙ 73 ∙ 137
──────── a Carmichael number:  41 ∙ 101 ∙ 461
──────── a Carmichael number:  41 ∙ 241 ∙ 521
──────── a Carmichael number:  41 ∙ 881 ∙ 12041
──────── a Carmichael number:  41 ∙ 1721 ∙ 35281

──────── a Carmichael number:  43 ∙ 127 ∙ 211
──────── a Carmichael number:  43 ∙ 211 ∙ 337
──────── a Carmichael number:  43 ∙ 271 ∙ 5827
──────── a Carmichael number:  43 ∙ 433 ∙ 643
──────── a Carmichael number:  43 ∙ 547 ∙ 673
──────── a Carmichael number:  43 ∙ 631 ∙ 1597
──────── a Carmichael number:  43 ∙ 3361 ∙ 3907

──────── a Carmichael number:  47 ∙ 1151 ∙ 1933
──────── a Carmichael number:  47 ∙ 3359 ∙ 6073
──────── a Carmichael number:  47 ∙ 3727 ∙ 5153

──────── a Carmichael number:  53 ∙ 79 ∙ 599
──────── a Carmichael number:  53 ∙ 157 ∙ 521

──────── a Carmichael number:  59 ∙ 1451 ∙ 2089

──────── a Carmichael number:  61 ∙ 181 ∙ 1381
──────── a Carmichael number:  61 ∙ 241 ∙ 421
──────── a Carmichael number:  61 ∙ 271 ∙ 571
──────── a Carmichael number:  61 ∙ 277 ∙ 2113
──────── a Carmichael number:  61 ∙ 421 ∙ 12841
──────── a Carmichael number:  61 ∙ 541 ∙ 3001
──────── a Carmichael number:  61 ∙ 661 ∙ 2521
──────── a Carmichael number:  61 ∙ 1301 ∙ 19841
──────── a Carmichael number:  61 ∙ 3361 ∙ 4021


69  Carmichael numbers found.

Ruby

Works with: Ruby version 1.9

<lang ruby># Generate Charmichael Numbers

require 'prime'

Prime.each(61) do |p|

 (2...p).each do |h3|
   g = h3 + p
   (1...g).each do |d|
     next if (g*(p-1)) % d != 0 or (-p*p) % h3 != d % h3
     q = 1 + ((p - 1) * g / d)
     next unless q.prime?
     r = 1 + (p * q / h3)
     next unless r.prime? and (q * r) % (p - 1) == 1
     puts "#{p} x #{q} x #{r}" 
   end
 end
 puts

end</lang>

Output:

3 x 11 x 17

5 x 29 x 73
5 x 17 x 29
5 x 13 x 17

7 x 19 x 67
7 x 31 x 73
7 x 13 x 31
7 x 23 x 41
7 x 73 x 103
7 x 13 x 19


13 x 61 x 397
13 x 37 x 241
13 x 97 x 421
13 x 37 x 97
13 x 37 x 61

17 x 41 x 233
17 x 353 x 1201

19 x 43 x 409
19 x 199 x 271

23 x 199 x 353

29 x 113 x 1093
29 x 197 x 953

31 x 991 x 15361
31 x 61 x 631
31 x 151 x 1171
31 x 61 x 271
31 x 61 x 211
31 x 271 x 601
31 x 181 x 331

37 x 109 x 2017
37 x 73 x 541
37 x 613 x 1621
37 x 73 x 181
37 x 73 x 109

41 x 1721 x 35281
41 x 881 x 12041
41 x 101 x 461
41 x 241 x 761
41 x 241 x 521
41 x 73 x 137
41 x 61 x 101

43 x 631 x 13567
43 x 271 x 5827
43 x 127 x 2731
43 x 127 x 1093
43 x 211 x 757
43 x 631 x 1597
43 x 127 x 211
43 x 211 x 337
43 x 433 x 643
43 x 547 x 673
43 x 3361 x 3907

47 x 3359 x 6073
47 x 1151 x 1933
47 x 3727 x 5153

53 x 157 x 2081
53 x 79 x 599
53 x 157 x 521

59 x 1451 x 2089

61 x 421 x 12841
61 x 181 x 5521
61 x 1301 x 19841
61 x 277 x 2113
61 x 181 x 1381
61 x 541 x 3001
61 x 661 x 2521
61 x 271 x 571
61 x 241 x 421
61 x 3361 x 4021

Seed7

The function isPrime below is borrowed from the Seed7 algorithm collection.

<lang seed7>$ include "seed7_05.s7i";

const func boolean: isPrime (in integer: number) is func

 result
   var boolean: prime is FALSE;
 local
   var integer: upTo is 0;
   var integer: testNum is 3;
 begin
   if number = 2 then
     prime := TRUE;
   elsif odd(number) and number > 2 then
     upTo := sqrt(number);
     while number rem testNum <> 0 and testNum <= upTo do
       testNum +:= 2;
     end while;
     prime := testNum > upTo;
   end if;
 end func;

const proc: main is func

 local
   var integer: p1 is 0;
   var integer: h3 is 0;
   var integer: g is 0;
   var integer: d is 0;
   var integer: p2 is 0;
   var integer: p3 is 0;
 begin
   for p1 range 2 to 61 do
     if isPrime(p1) then
       for h3 range 2 to p1 do
         g := h3 + p1;
         for d range 1 to pred(g) do
           if (g * pred(p1)) mod d = 0 and -p1 ** 2 mod h3 = d mod h3 then
             p2 := 1 + pred(p1) * g div d;
             if isPrime(p2) then
               p3 := 1 + p1 * p2 div h3;
               if isPrime(p3) and (p2 * p3) mod pred(p1) = 1 then
                 writeln(p1 <& " * " <& p2 <& " * " <& p3 <& " = " <& p1*p2*p3);
               end if;
             end if;
           end if;
         end for;
       end for;
     end if;
   end for;
 end func;</lang>

Output:

3 * 11 * 17 = 561
5 * 29 * 73 = 10585
5 * 17 * 29 = 2465
5 * 13 * 17 = 1105
7 * 19 * 67 = 8911
7 * 31 * 73 = 15841
7 * 13 * 31 = 2821
7 * 23 * 41 = 6601
7 * 73 * 103 = 52633
7 * 13 * 19 = 1729
13 * 61 * 397 = 314821
13 * 37 * 241 = 115921
13 * 97 * 421 = 530881
13 * 37 * 97 = 46657
13 * 37 * 61 = 29341
17 * 41 * 233 = 162401
17 * 353 * 1201 = 7207201
19 * 43 * 409 = 334153
19 * 199 * 271 = 1024651
23 * 199 * 353 = 1615681
29 * 113 * 1093 = 3581761
29 * 197 * 953 = 5444489
31 * 991 * 15361 = 471905281
31 * 61 * 631 = 1193221
31 * 151 * 1171 = 5481451
31 * 61 * 271 = 512461
31 * 61 * 211 = 399001
31 * 271 * 601 = 5049001
31 * 181 * 331 = 1857241
37 * 109 * 2017 = 8134561
37 * 73 * 541 = 1461241
37 * 613 * 1621 = 36765901
37 * 73 * 181 = 488881
37 * 73 * 109 = 294409
41 * 1721 * 35281 = 2489462641
41 * 881 * 12041 = 434932961                                                                                                                                                 
41 * 101 * 461 = 1909001                                                                                                                                                     
41 * 241 * 761 = 7519441                                                                                                                                                     
41 * 241 * 521 = 5148001                                                                                                                                                     
41 * 73 * 137 = 410041                                                                                                                                                       
41 * 61 * 101 = 252601                                                                                                                                                       
43 * 631 * 13567 = 368113411                                                                                                                                                 
43 * 271 * 5827 = 67902031                                                                                                                                                   
43 * 127 * 2731 = 14913991                                                                                                                                                   
43 * 127 * 1093 = 5968873                                                                                                                                                    
43 * 211 * 757 = 6868261                                                                                                                                                     
43 * 631 * 1597 = 43331401                                                                                                                                                   
43 * 127 * 211 = 1152271
43 * 211 * 337 = 3057601
43 * 433 * 643 = 11972017
43 * 547 * 673 = 15829633
43 * 3361 * 3907 = 564651361
47 * 3359 * 6073 = 958762729
47 * 1151 * 1933 = 104569501
47 * 3727 * 5153 = 902645857
53 * 157 * 2081 = 17316001
53 * 79 * 599 = 2508013
53 * 157 * 521 = 4335241
59 * 1451 * 2089 = 178837201
61 * 421 * 12841 = 329769721
61 * 181 * 5521 = 60957361
61 * 1301 * 19841 = 1574601601
61 * 277 * 2113 = 35703361
61 * 181 * 1381 = 15247621
61 * 541 * 3001 = 99036001
61 * 661 * 2521 = 101649241
61 * 271 * 571 = 9439201
61 * 241 * 421 = 6189121
61 * 3361 * 4021 = 824389441

Sidef

Translation of: Perl

<lang ruby>var ntheory = frequire('ntheory');

ntheory.forprimes({ |*p|

  p = Number.new(p[-1]);
  range(2, p-1).each { |h3|
     var ph3 = (p + h3);
     range(1, ph3-1).each { |d|
        ((-p * p) % h3) != (d % h3) && next;
        ((p-1)*ph3) % d && next;
        var q = 1+((p-1) * ph3 / d);
        ntheory.is_prime(q) || next;
        var r = 1+((p*q - 1)/h3);
        ntheory.is_prime(r) || next;
        (q*r) % (p-1) == 1 || next;
        printf("%2d x %5d x %5d = %s\n",p,q,r,ntheory.vecprod(p,q,r));
     }
  }

}, 3, 61);</lang>

Output:

 3 x    11 x    17 = 561
 5 x    29 x    73 = 10585
 5 x    17 x    29 = 2465
 5 x    13 x    17 = 1105
 ... full output is 69 lines ...
61 x   661 x  2521 = 101649241
61 x   271 x   571 = 9439201
61 x   241 x   421 = 6189121
61 x  3361 x  4021 = 824389441

Tcl

Using the primality tester from the Miller-Rabin task... <lang tcl>proc carmichael {limit {rounds 10}} {

   set carmichaels {}
   for {set p1 2} {$p1 <= $limit} {incr p1} {

if {![miller_rabin $p1 $rounds]} continue for {set h3 2} {$h3 < $p1} {incr h3} { set g [expr {$h3 + $p1}] for {set d 1} {$d < $h3+$p1} {incr d} { if {(($h3+$p1)*($p1-1))%$d != 0} continue if {(-($p1**2))%$h3 != $d%$h3} continue

set p2 [expr {1 + ($p1-1)*$g/$d}] if {![miller_rabin $p2 $rounds]} continue

set p3 [expr {1 + $p1*$p2/$h3}] if {![miller_rabin $p3 $rounds]} continue

if {($p2*$p3)%($p1-1) != 1} continue lappend carmichaels $p1 $p2 $p3 [expr {$p1*$p2*$p3}] } }

   }
   return $carmichaels

}</lang> Demonstrating: <lang tcl>set results [carmichael 61 2] puts "[expr {[llength $results]/4}] Carmichael numbers found" foreach {p1 p2 p3 c} $results {

   puts "$p1 x $p2 x $p3 = $c"

}</lang>

Output:

69 Carmichael numbers found
3 x 11 x 17 = 561
5 x 29 x 73 = 10585
5 x 17 x 29 = 2465
5 x 13 x 17 = 1105
7 x 19 x 67 = 8911
7 x 31 x 73 = 15841
7 x 13 x 31 = 2821
7 x 23 x 41 = 6601
7 x 73 x 103 = 52633
7 x 13 x 19 = 1729
13 x 61 x 397 = 314821
13 x 37 x 241 = 115921
13 x 97 x 421 = 530881
13 x 37 x 97 = 46657
13 x 37 x 61 = 29341
17 x 41 x 233 = 162401
17 x 353 x 1201 = 7207201
19 x 43 x 409 = 334153
19 x 199 x 271 = 1024651
23 x 199 x 353 = 1615681
29 x 113 x 1093 = 3581761
29 x 197 x 953 = 5444489
31 x 991 x 15361 = 471905281
31 x 61 x 631 = 1193221
31 x 151 x 1171 = 5481451
31 x 61 x 271 = 512461
31 x 61 x 211 = 399001
31 x 271 x 601 = 5049001
31 x 181 x 331 = 1857241
37 x 109 x 2017 = 8134561
37 x 73 x 541 = 1461241
37 x 613 x 1621 = 36765901
37 x 73 x 181 = 488881
37 x 73 x 109 = 294409
41 x 1721 x 35281 = 2489462641
41 x 881 x 12041 = 434932961
41 x 101 x 461 = 1909001
41 x 241 x 761 = 7519441
41 x 241 x 521 = 5148001
41 x 73 x 137 = 410041
41 x 61 x 101 = 252601
43 x 631 x 13567 = 368113411
43 x 271 x 5827 = 67902031
43 x 127 x 2731 = 14913991
43 x 127 x 1093 = 5968873
43 x 211 x 757 = 6868261
43 x 631 x 1597 = 43331401
43 x 127 x 211 = 1152271
43 x 211 x 337 = 3057601
43 x 433 x 643 = 11972017
43 x 547 x 673 = 15829633
43 x 3361 x 3907 = 564651361
47 x 3359 x 6073 = 958762729
47 x 1151 x 1933 = 104569501
47 x 3727 x 5153 = 902645857
53 x 157 x 2081 = 17316001
53 x 79 x 599 = 2508013
53 x 157 x 521 = 4335241
59 x 1451 x 2089 = 178837201
61 x 421 x 12841 = 329769721
61 x 181 x 5521 = 60957361
61 x 1301 x 19841 = 1574601601
61 x 277 x 2113 = 35703361
61 x 181 x 1381 = 15247621
61 x 541 x 3001 = 99036001
61 x 661 x 2521 = 101649241
61 x 271 x 571 = 9439201
61 x 241 x 421 = 6189121
61 x 3361 x 4021 = 824389441

zkl

Using the Miller-Rabin primality test in lib GMP. <lang zkl>var BN=Import("zklBigNum"), bi=BN(0); // gonna recycle bi primes:=T(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61); var p2,p3; cs:=[[(p1,h3,d); primes; { [2..p1 - 1] }; // list comprehension

     { [1..h3 + p1 - 1] },

{ ((h3 + p1)*(p1 - 1)%d == 0 and ((-p1*p1):mod(_,h3) == d%h3)) },//guard { (p2=1 + (p1 - 1)*(h3 + p1)/d):bi.set(_).probablyPrime() },//guard { (p3=1 + (p1*p2/h3)):bi.set(_).probablyPrime() }, //guard { 1==(p2*p3)%(p1 - 1) }; //guard

  { T(p1,p2,p3) }  // return list of three primes in Carmichael number

]]; fcn mod(a,b) { m:=a%b; if(m<0) m+b else m }</lang> <lang>cs.len().println(" Carmichael numbers found:"); cs.pump(Console.println,fcn([(p1,p2,p3)]){

  "%2d * %4d * %5d = %d".fmt(p1,p2,p3,p1*p2*p3) });</lang>

Output:

69 Carmichael numbers found:
 3 *   11 *    17 = 561
 5 *   29 *    73 = 10585
 5 *   17 *    29 = 2465
 5 *   13 *    17 = 1105
 7 *   19 *    67 = 8911
...
61 *  181 *  1381 = 15247621
61 *  541 *  3001 = 99036001
61 *  661 *  2521 = 101649241
61 *  271 *   571 = 9439201
61 *  241 *   421 = 6189121
61 * 3361 *  4021 = 824389441