Does a Carmichael number with $k$ prime factors exist for every $k\ge 3$?

Question

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 $k\ge 3$ ?

Answer 1

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.