Problem

Source: 2021 Taiwan TST Round 3 Independent Study 2-N

Tags: number theory, Taiwan



Let $n$ be a given positive integer. We say that a positive integer $m$ is $n$-good if and only if there are at most $2n$ distinct primes $p$ satisfying $p^2\mid m$. (a) Show that if two positive integers $a,b$ are coprime, then there exist positive integers $x,y$ so that $ax^n+by^n$ is $n$-good. (b) Show that for any $k$ positive integers $a_1,\ldots,a_k$ satisfying $\gcd(a_1,\ldots,a_k)=1$, there exist positive integers $x_1,\ldots,x_k$ so that $a_1x_1^n+a_2x_2^n+\cdots+a_kx_k^n$ is $n$-good. (Remark: $a_1,\ldots,a_k$ are not necessarily pairwise distinct) Proposed by usjl.