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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.

Link : Can a carmichael number have consecutive prime factors?

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

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

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