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Number theory on Gaussian integers at RMM almost

Source: RMM 2023 Shortlist N1

February 29, 2024
number theorygaussiangaussian integerDivisibilityRMM Shortlist

Problem Statement

Let nn be a positive integer. Let SS be a set of ordered pairs (x,y)(x, y) such that 1≤x≤n1\leq x \leq n and 0≤y≤n0 \leq y \leq n in each pair, and there are no pairs (a,b)(a, b) and (c,d)(c, d) of different elements in SS such that a2+b2a^2+b^2 divides both ac+bdac+bd and ad−bcad - bc. In terms of nn, determine the size of the largest possible set SS.
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