On the prime factors of the number 2p-1 - 1

Abstract

From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes p in the arithmetical progression ax + b (x = 0, 1, 2,…), where a and b are relatively prime positive integers, such that the number 2p−1 − 1 has at least k composite factors of the form (p − 1)x + 1. The following question arises:For any given natural number k, do there exist infinitely many primes p such that the number 2p−1 − 1 has k prime factors of the form(p − 1)x + 1 and p ≡ b (mod a), where a and b are coprime positive integers?