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2023 SSMO Speed Round Problems/Problem 6

Problem

Find the smallest odd prime that does not divide $2^{75!} - 1$.

Solution

Let this odd prime be $p$.

Note that $2^{75!} - 1$ is divisible by $p$ if \[2^{75!} \equiv 1 \pmod{p}\] or $p - 1 \mid 75!$.

As such, $p$ is the smallest prime of the form $2q + 1$ where $q > 75$ is also prime.

After testing some of the primes above 75, we find that $q=68$ is the smallest prime, meaning the answer is $\boxed{167}.$

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