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xy^3z + 2x^3z^3-3x^5y = 0 has odd no of integer solutions in a set

Source: 2013 Swedish Mathematical Competition p1

April 30, 2021
number theorydiophantineDiophantine equation

Problem Statement

For r>0r> 0 denote by BrB_r the set of points at distance at most rr length units from the origin. If PrP_r is the set of the points in BrB_r whit integer coordinates, show that the equation xy3z+2x3z3−3x5y=0xy^3z + 2x^3z^3-3x^5y = 0 has an odd number of solutions (x,y,z)(x, y, z) in PrP_r.
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