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10th EMC - Factorials and perfect squares

Source: 10th European Mathematical Cup - Problem J3

December 22, 2021
factorialPerfect Squarenumber theoryemcEuropean Mathematical Cup

Problem Statement

Let ℓ\ell be a positive integer. We say that a positive integer kk is nice if k!+ℓk!+\ell is a square of an integer. Prove that for every positive integer n⩾ℓn \geqslant \ell, the set {1,2,…,n2}\{1, 2, \ldots,n^2\} contains at most n2−n+ℓn^2-n +\ell nice integers. \\ \\ (Théo Lenoir)
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