Problem

Source: XII International Festival of Young Mathematicians Sozopol 2023, Theme for 10-12 grade

Tags: number theory



Does there exist a natural number $n \geq 2$ such that: a) $\frac{2^{n-1}+1}{n}$ is a natural number? b) $\frac{2^{2n-1}-1}{n}$ is a prime number?