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NT with condition that is expected to be almost always true by PNT

Source: 2024 CTST P16

March 25, 2024
number theory

Problem Statement

m>1m>1 is an integer such that [2m−m+1,2m][2m-\sqrt{m}+1, 2m] contains a prime. Prove that for any pairwise distinct positive integers a1a_1, a2a_2, …\dots, ama_m, there is always 1≤i,j≤m1\leq i,j\leq m such that ai(ai,aj)≥m\frac{a_i}{(a_i, a_j)}\geq m.
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