# Is it known whether the possible number of prime factors of a Carmichael-number is bounded?

Let $n(C)$ be the number of prime factors of the Carmichael-number $C$.

I Conjecture $\lim sup_{C\rightarrow \infty} n(C)=\infty$

In other words, the sequence $n(C)$, $C$ running over the Carmichael-numbers, is unbounded.

I learnt that Dickson's conjecture implies that this is the case. There are arbitary long strictly increasing vectors $v_1,...,v_n$ ($n\ge 3$) with $\sum_{j=1}^n \frac{1}{v_j}=1$. If $L\ :=\ lcm(v_1,...,v_n)$, then $\prod_{j=1}^n (\frac{L^2}{x_j}\times m+1)$ is a Carmichael-number if $\frac{L^2}{x_j}\times m+1$ is prime for $j=1,...,n$, and Dickson's conjecture implies that such a number $m$ always exists.

I also learnt that it is not known, whether there are infinite many Carmichael numbers with $k$ prime factors for any fixed number $k\ge 3$. But maybe my conjecture can be proven (or disproven).

• A vote up for your conjecture, good luck! – user243301 Nov 3 '15 at 19:52