On the Diophantine equations \(z^2=f(x)^2\pm f(y)^2\) involving quartic polynomials (Q2322532)

The authors apply classical elliptic curve theory and elementary computations to show that for \(f(X) = (X^3-X)(X + \frac{1-k^2}{2k})\) (with \(k \neq 0, \pm 1\)) the equation \(z^2 = f(x)^2 + f(y)^2\) has rational solutions, and similarly for \(z^2 = f(x)^2 - f(y)^2\) when the final factor of \(f(X)\) above is changed to \(X - \frac{2k}{k^2+1}\). This answers, in the setting of quartic polynomials, Question 4.3 of \textit{M. Ulas} and \textit{A. Togbé} [Publ. Math. 76, No. 1--2, 183--201 (2010); Zbl 1218.11034].