Bachet equations and cubic resolvents (Q2868484)

In the paper under review, the author relates the problem of finding rational solutions of the diophantine equation \(Y^2=X^3+k, k\in\mathbb{Z}\) to the question concerning the existence of a rational root of the equation \(f(X)=0\), where \(f(X)=X^3-b^2X^2+k\). The latter question is resolved by characterizing these quartic polynomials that become biquadratic by a change of a variable whose cubic resolvent is the polynomial \(f\). As an application of the results the author proves that the equation \(Y^2=X^3+k\) has a rational solution if and only if there are \(a, b\in\mathbb{Q}\) such that \(k=-a^2(a-b^2)\).