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> $N=2^p -1$, $p$ prime $\implies$ $2^{N-1} \equiv 1$ mod $N$
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I thought about using Euler's Theorem: clearly $(2, 2^p -1)=1$, so $2^{\varphi (N)} \equiv 1$ mod $N$, which will finish the proof iff N is a prime.

Uh oh, $2^{11} -1$ is not prime, so this argument won't work.