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2022 SSMO Speed Round Problems/Problem 6

Revision as of 12:59, 3 July 2023 by Pinkpig (talk | contribs) (Created page with "==Problem== Find the smallest odd prime that does not divide <math>2^{75!} - 1</math>. ==Solution== Let this odd prime be <math>p</math>. Note that <math>2^{75!} - 1</math>...")
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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.

This is called a \textit{safe} prime in literature and checking that $\boxed{167}$ is the first such $p$.

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