Question 17
Let $n$ be a given positive integer. A positive integer $m$ is called $n$-good if at most $2n$ distinct primes $p$ satisfy $p^2\mid m$.
Part (a). Prove that for any two coprime positive integers $a,b$, there exist positive integers $x,y$ such that $ax^n+by^n$ is $n$-good.
Part (b). Prove that for any $k$ positive integers $a_1,\dots,a_k$ with $\gcd(a_1,\dots,a_k)=1$, there exist positive integers $x_1,\dots,x_k$ such that $a_1x_1^n+\cdots+a_kx_k^n$ is $n$-good. The numbers $a_1,\dots,a_k$ need not be pairwise distinct.
Part (a). Prove that for any two coprime positive integers $a,b$, there exist positive integers $x,y$ such that $ax^n+by^n$ is $n$-good.
Part (b). Prove that for any $k$ positive integers $a_1,\dots,a_k$ with $\gcd(a_1,\dots,a_k)=1$, there exist positive integers $x_1,\dots,x_k$ such that $a_1x_1^n+\cdots+a_kx_k^n$ is $n$-good. The numbers $a_1,\dots,a_k$ need not be pairwise distinct.