Problem

Source: Turkey TST 2013 - Day 1 - P1

Tags: induction, number theory, relatively prime, number theory proposed



Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]