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(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n, smallest n such no of ordered pairs d(n)=61

Source: 2020 Czech-Polish-Slovak Match p5

October 8, 2020
number theoryDiophantine equation

Problem Statement

Let nn be a positive integer and let d(n)d(n) denote the number of ordered pairs of positive integers (x,y)(x,y) such that (x+1)2−xy(2x−xy+2y)+(y+1)2=n(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n. Find the smallest positive integer nn satisfying d(n)=61d(n) = 61.
(Patrik Bak, Slovakia)
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