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.