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Let $N$ be a Carmichael number , $p$ its smallest prime factor, $q$ its largest prime factor.

Which is the largest possible prime $p$, if the range $[p,q]$ only contains at most $20$ primes ?

The Carmichael number $43\times67\times89\times103\times127\times137$ has this property because there are $20$ primes in the range $[43,137]$. I did not find an example with $p>43$.

Can it be shown, that $p>43$ is impossible ?

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