# How small can $\frac{p_n}{p_1}$ become, if $\prod_{j=1}^n p_j$ is a Carmichael-number?

Suppose, $N=\prod_{j=1}^n p_j$ is a Carmichael-number. Define $c(N)=\frac{p_n}{p_1}$. The numbers $p_1,...,p_n$ are the prime factors of the carmichael-number $N$. $c(N)$ is the ratio of the largest and the smallest prime factor.

What is the minimal possible value for $c(N)$ ?

I do not even know, if there is a minimum, or if $c(N)$ can be arbitary close to $1$.

The minimum values I found so far are $c(30241 \times 32257 \times 34273)=1.13333$ for $3$ factors and $c(2381\times2521\times2549\times3529)=1.48215$ for $4$ factors.

New record : $c(521137\times 530443 \times 567667)=1.08929$

• If a minimal value can be proven, this would be a great step towards solving the linked question (in the negative sense). – Peter Oct 20 '15 at 19:06