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.



Task

Find Carmichael numbers of the form:

  Prime1 × Prime2 × Prime3 
where    (Prime1 < Prime2 < Prime3)    for all    Prime1    up to   61.

(See page 7 of   Notes by G.J.O Jameson March 2010   for solutions.)



Pseudocode

For a given   



P
r
i
m

e

1




{\displaystyle 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



related task
Chernick's Carmichael numbers

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;