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FACTORS OF CARMICHAEL NUMBERS AND AN EVEN WEAKER
$k$-TUPLES CONJECTURE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 20 February 2019, pp. 376-384
- Print publication:
- June 2019
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- Article
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- You have access
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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.
