Carmichael Numbers with (p+1) | (n-1)

Hugh Williams posed an interesting problem of whether there exists a Carmichael number N with p+1|N+1 for all primes p|N. Othman Echi calls such numbers Williams numbers (more precisely, 1-Williams numbers). Carl Pomerance gave a heuristic argument that there are infinitely many counterexamples to the Baillie–PSW probable prime test. Based on some reasonable assumptions there exist infinitely many Williams numbers. There are no examples less than 264≈2×1019. Williams proved that any such numbers must have more than three prime factors. In this paper we prove that there are only finitely many Williams numbers N=∏i=1dpi with a given set of d−3 prime factors p1,…,pd−3. Several methods for the organization of a search for Williams numbers are given. We report that if there are any Williams numbers with exactly four prime factors, then the smallest prime factor is greater than 2×104.