Prove that for every prime $p$ exists infinitely many integers $n$ such that $p \mid 2^n-n$.
I have no idea how to prove that.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this communityProve that for every prime $p$ exists infinitely many integers $n$ such that $p \mid 2^n-n$.
I have no idea how to prove that.
Let $o:=\operatorname{ord}(2,p)$ be the smallest positive number with $2^o\equiv 1\pmod p$
Then we have for every natural number $k$ : $2^{ok}\equiv 1\pmod p$
Because of $1 < o < p$ there exists $q$ with $oq\equiv 1\pmod p$
So we have $2^{oq}\equiv oq\equiv 1\pmod p$