Problem

Source: Peruvian IMO TST 2020 P1

Tags: Peru, TST, number theory, Perfect Powers, prime numbers



Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$