Is there an efficient method to construct random Carmichael-numbers with $50-100$ prime factors ?
The method with vectors $p_1,...,p_k$ , where $\frac{1}{p_1}+...\frac{1}{p_k}=1$ which gives a formula like $(6n+1)(12n+1)(18n+1)$ , which is a Carmichael number, if all the factors are prime, is useless because it is too difficult to find a number $n$, such that so many factors are simultanoeusly prime.
Erdoes method produces Carmichael numbers with not too many factors relatively easy, but it would take rather long to find a subset of $50-100$ primes with $\prod_{j=1}^k p_j\equiv 1\ (\ mod \ L)$
Do the quadratic residues help ? Which method is best to construct my own Carmichael-"monster-numbers" ?