2
$\begingroup$

Here

https://oeis.org/search?q=carmichael+number+factors&language=german&go=Suche

the smallest Carmichael numbers with $k=3,...,35$ prime factors are shown. In Wikipdia, it is stated, that a Carmichael number with over $1,000,000$ prime factors has been constructed.

Is there a Carmichael number with $k$ prime factors for every $k\ge 3$ ?

$\endgroup$
  • $\begingroup$ The Wikipedia article says a Carmichael number with over 16 million digits and $1,101,518$ factors was found. It also states that there are at least $n^{2/7}$ Carmichael numbers less than $n$ I would expect that this means there are Carmichael numbers with arbitrarily many factors. That certainly leaves open the possibility that there is some number of factors not represented. $\endgroup$ – Ross Millikan Oct 15 '15 at 16:22
2
$\begingroup$

This is an open problem, as mentioned in Richard Guy's Unsolved Problems in Number Theory, 2004 edition, as well as the paper "Carmichael numbers with three prime factors" by Heath-Brown in 2007. An even more recent paper that mentions its status as open is "Constructing Carmichael numbers through improved subset-product algorithms" by Alford, Grantham, Hayman, and Shallue, in 2013.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.