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Does there exist a prime number $p$ such that $p^2 \mid 2^{p-1}-1$ ?

I tried for some small number $p$ and I think that it does, but I don't know how to prove this.

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Actually there is.

$1093$ does the job.

These primes are named Wieferich primes but we don't know if there are infinitely many.
For more see here

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