Some number theory questions

1. For all n > 1, n does not divide $2^n - 1$.
   Proof: Suppose not. Then n is odd as $2^n - 1$ is odd. Let $p$ be the smallest prime dividing n. We have $(p - 1, n) = 1$. By division algorithm, $n = q(p - 1) +k$ for some integers q and k with $0 < k < p$. By Fermat’s Little Theorem, $2^{p - 1} \equiv 1$ (mod p), so $2^k \equiv (2^{p - 1})^q2^k \equiv 2^n \equiv 1$ (mod p). This results in a contradiction by infinite descent.

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