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

#!/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