We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
$k$-TUPLES CONJECTURE
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.
