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mn and (m + 1)(n + 1) are perfect squares , nice pair of pos. integers

Source: Tournament of Towns, Junior A-Level , Fall 2019 p5

April 20, 2020
Perfect SquaresPerfect Squarenumber theory

Problem Statement

Let us say that the pair (m,n)(m, n) of two positive different integers m and n is nice if mnmn and (m+1)(n+1)(m + 1)(n + 1) are perfect squares. Prove that for each positive integer m there exists at least one n>mn > m such that the pair (m,n)(m, n) is nice.
(Yury Markelov)
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