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Omopeiro numbers

Source: Lusophon Mathematical Olympiad 2021 Problem 6

December 19, 2021
number theorycombinatorics

Problem Statement

A positive integer nn is called omopeiroomopeiro if there exists nn non-zero integers that are not necessarily distinct such that 20212021 is the sum of the squares of those nn integers. For example, the number 22 is not an omopeiroomopeiro, because 20212021 is not a sum of two non-zero squares, but 20212021 is an omopeiroomopeiro, because 2021=12+12+⋯+122021=1^2+1^2+ \dots +1^2, which is a sum of 20212021 squares of the number 11.
Prove that there exist more than 1500 omopeiroomopeiro numbers.
Note: proving that there exist at least 500 omopeiroomopeiro numbers is worth 2 points.
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