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Represent n^4-k^4 as sum of (a+b)(a^2+b^2)

Source: Ukrainian Mathematical Olympiad 2021. Day 2, Problem 9.5

December 21, 2023
number theory

Problem Statement

Let MM denote the set consisting of positive integers that can be represented as (a+b)(a2+b2)(a + b)(a^2 + b^2), where a,ba, b are some distinct positive integers. Prove that for any positive integer n>kn > k, the number n4−k4n^4 - k^4 is the sum of some n−kn - k distinct elements of the set MM.
Proposed by Oleksii Masalitin
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