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A problem with perfect powers

Source: P6, Mexico Center Regional Olympiad 2019

November 28, 2019

number theory

Problem Statement

Find all positive integers

a_{2019}

with the next property: If

d

is a positive integer less or equal to

m

and it isn't coprime to

m

, then there exist positive integers

a_{1}, a_{2}

,. . .,

a_{2019}

(where all of them are coprimes to

m

) such that

m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}

is a perfect power.

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