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3y^4 + 4cy^3 + 2xy + 48 = 0

Source: Austrian - Polish 1995 APMC

May 3, 2020
number theorysquarefreePerfect SquarediophantineDiophantine equation

Problem Statement

Consider the equation 3y4+4cy3+2xy+48=03y^4 + 4cy^3 + 2xy + 48 = 0, where cc is an integer parameter. Determine all values of cc for which the number of integral solutions (x,y)(x,y) satisfying the conditions (i) and (ii) is maximal: (i) ∣x∣|x| is a square of an integer; (ii) yy is a squarefree number.
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