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All four gcds are at least c*sqrt(n)

Source: Miklós Schweitzer 2019, Problem 3

December 27, 2019
number theory

Problem Statement

Prove that there are infinitely many integers mm, nn, such that 1<m<n1 < m < n, and the greatest common divisors (m,n)(m, n), (m,n+1)(m, n+1), (m+1,n)(m+1, n) and (m+1,n+1)(m+1, n+1) are all greater than n/999\sqrt{n}/999.
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