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

import "/fmt" for Fmt
import "/math" for Int
 
var mod = Fn.new { |n, m| ((n%m) + m) % m }
 
var carmichael = Fn.new { |p1|
    for (h3 in 2...p1) {
        for (d in 1...h3 + p1) {
            if ((h3 + p1) * (p1 - 1) % d == 0 && mod.call(-p1 * p1, h3) == d % h3) {
                var p2 = 1 + ((p1 - 1) * (h3 + p1) / d).floor
                if (Int.isPrime(p2)) {
                    var p3 = 1 + (p1 * p2 / h3).floor
                    if (Int.isPrime(p3)) {
                        if (p2 * p3 % (p1 - 1) == 1) {
                            var c = p1 * p2 * p3
                            Fmt.print("$2d   $4d   $5d  $10d", p1, p2, p3, c)
                        }
                    }
                }
            }
        }
    }
}
 
System.print("The following are Carmichael munbers for p1 <= 61:\n")
System.print("p1     p2      p3     product")
System.print("==     ==      ==     =======")
for (p1 in 2..61) {
    if (Int.isPrime(p1)) carmichael.call(p1)
}