FACTORS OF CARMICHAEL NUMBERS AND AN EVEN WEAKER k-TUPLES CONJECTURE

One of the open questions in the study of Carmichael numbers is whether, for a given R >= 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.