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Diophantine equation has too many solutions for some n

Source: Greece National Olympiad 2024, Problem 4

February 24, 2024
number theory

Problem Statement

Prove that there exists an integer n≥1n \geq 1, such that number of all pairs (a,b)(a, b) of positive integers, satisfying 1a−b−1a+1b=1n\frac{1}{a-b}-\frac{1}{a}+\frac{1}{b}=\frac{1}{n} exceeds 2024.2024.
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