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

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}";
            }
        }
    }
}