Carmichael Numbers With Three Prime Factors

A Carmichael number (or absolute pseudo-prime) is a composite positive integer n such that n|an − a for every integer a. It is not difficult to prove that such an integer must be square-free, with at least 3 prime factors. Moreover if the numbers p = 6m + 1, q = 12m + 1 and r = 18m + 1 are all prime, then n = pqr will be a Carmichael number. However it is not currently known whether there are infinitely many prime triplets of this form. Indeed it is not known whether or not there are infinitely many Carmichael numbers with 3 prime factors. None the less it was proved by Alford, Granville and Pomerance [1] that there are infinitely many Carmichael numbers. Let C3(x) denote the number of Carmichael numbers n ≤ x having ω(n) = 3. It has been conjectured by Granville and Pomerance [3] that