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Classical NT about density of numbers of a special form

Source: Bulgaria MO Regional round 2024, 12.4

February 13, 2024

number theory

Problem Statement

Find all pairs of positive integers

(n, k)

such that all sufficiently large odd positive integers

m

are representable as

m=a_1^{n^2}+a_2^{(n+1)^2}+\ldots+a_k^{(n+k-1)^2}+a_{k+1}^{(n+k)^2}

for some non-negative integers

a_1, a_2, \ldots, a_{k+1}