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Problem 4 of Second round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

September 9, 2022
number theory

Problem Statement

a) Prove that for each positive integer nn the number or ordered pairs of integers (x,y)(x,y) for which x2−xy+y2=nx^2-xy+y^2=n is finite and is multiple of 6. b) Find all ordered pairs of integers (x,y)(x,y) for which x2−xy+y2=727x^2-xy+y^2=727.
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