MathDB
f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3)

Source: 1998 Belarus TST 4.3

December 25, 2020
number theoryPerfect Squareperfect cube

Problem Statement

a) Let f(x,y)=x3+(3y2+1)x2+(3y4−y2+4y−1)x+(y6−y4+2y3)f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3). Prove that if for some positive integers a,ba, b the number f(a,b)f(a, b) is a cube of an integer then f(a,b)f(a, b) is also a square of an integer.
b) Are there infinitely many pairs of positive integers (a,b)(a, b) for which f(a,b)f(a, b) is a square but not a cube ?
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