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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 $k\ge 3$ ?

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  • $\begingroup$ 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. $\endgroup$ Oct 15, 2015 at 16:22

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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.

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