MathDB

Let

n

be a given positive integer. We say that a positive integer

m

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

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

n

-good.
(Remark:

a_1,\ldots,a_k

are not necessarily pairwise distinct)
Proposed by usjl.

Problem Statement