1.) Show for every integer of this form: (6m + 1)(12m + 1)(18m + 1), with m being a pos. integer such that 6m + 1, 12m + 1, 18m + 1 are all primes, is a Carmichael number
2.) From part #1, show why 294409 = 37*73*109 is a Carmichael number
3.) Determine the least pos. residue of 765^(1472043) (mod 294409)
From my number thry book:
A composite integer n that satisfies b^(n-1) = 1 (mod n) for all pos. integers b with (b,n) = 1 is called a Carmichael number or an absolute pseudoprime.
And a THM about thsee numbers:
If n = q_1q_2...q_k, where q_j are distinct primes that satisfy (q_j - 1) | (n - 1) for all j and k > 2, then n is a Carmichael number.