Ivan Shishkin, Rye (1878)

A necessary condition to be prime ?

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Let $n$ be a positive integer.

Assume $2^n-1$ is prime, show that $n$ is also prime.
Assume now $2^{n}+1$ is prime, show that $n$ is a power of $2$ .
By computing the first terms and/or using a computer, show that there exists integers of the form $2^p-1, p$ prime or $2^{2^q}+1, q$ non-negative integer, that are not prime numbers.

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