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product (11x_i^2 +12y_i^2) is perfect square

Source: 2012 Argentina OMA Finals L3 p2

January 15, 2023
number theoryPerfect Square

Problem Statement

Determine all natural numbers nn for which there are 2n2n distinct positive integers x1,…,xn,y1,…,ynx_1,…,x_n,y_1,…,y_n such that the product (11x12+12y12)(11x22+12y22)…(11xn2+12yn2)(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n) is a perfect square.
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