FACTORS OF CARMICHAEL NUMBERS AND AN EVEN WEAKER -TUPLES CONJECTURE

Abstract

One of the open questions in the study of Carmichael numbers is whether, for a given $R\geq 3$ , there exist infinitely many Carmichael numbers with exactly $R$ prime factors. Chernick [‘On Fermat’s simple theorem’, Bull. Amer. Math. Soc. 45 (1935), 269–274] proved that Dickson’s $k$ -tuple conjecture would imply a positive result for all such $R$ . Wright [‘Factors of Carmichael numbers and a weak $k$ -tuples conjecture’, J. Aust. Math. Soc. 100(3) (2016), 421–429] showed that a weakened version of Dickson’s conjecture would imply that there are an infinitude of $R$ for which there are infinitely many such Carmichael numbers. In this paper, we improve on our 2016 result by weakening the required conjecture even further.