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NT equations make a huge comeback

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 11.1

March 19, 2024
greatest common divisorleast common multiplenumber theory

Problem Statement

Find all pairs a,ba, b of positive integers, for which
(a,b)+3[a,b]=a3āˆ’b3(a, b) + 3[a, b] = a^3 - b^3
Here (a,b)(a, b) denotes the greatest common divisor of a,ba, b, and [a,b][a, b] denotes the least common multiple of a,ba, b.
Proposed by Oleksiy Masalitin
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