I wonder what we do know about $2^p\ mod \ p^2\\$ where $p$ is a prime?
I know $2^p\equiv\ r*p+2\mod \ p^2$. Do we know anything about $r$?
What do we know about $2^p\ mod \ p^2$?
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number-theory
elementary-number-theory
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0The values of $r$ are given by a sequence in the Online Encyclopedia of Integer Sequences: http://oeis.org/A179077 . There's no indication there that anything is known. – 2017-01-30
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0See [Fermat quotient.](https://en.wikipedia.org/wiki/Fermat_quotient) and [$p-$derivation.](https://en.wikipedia.org/wiki/P-derivation) – 2017-01-30