On the equation \(Y^2=X^6+k\) (Q2888160)

Let \(k\in\mathbb{Z}\setminus\{0\}\) and consider the hyperelliptic curve \(C_{k}:\;y^2=x^6+k\). In the paper under review the authors are interested in characterization of the set of rational points on the curve \(C_{k}\) for all integers \(k\) in the range \(|k|\leq 50\). The genus of \(C_{k}\) is 2 and thus Faltings's theorem implies that the set of rational points on \(C_{k}\) is finite. Using variety of methods the authors were able to find all rational points on \(C_{k}\) in the considered range, expect for two values \(k=-47, -39\). The methods used in order to get the results start with elementary considerations (congruences and quadratic reciprocity law) through the algebraic number theory approaches (factorization and units over number fields) to the elliptic Chabauty techniques. The most difficult analysis was performed for \(k=15, 43, -11, -15\). In order to tackle these four cases the authors used a clever combination of all mentioned methods.NEWLINENEWLINEAs a challenge the authors propose to find all rational points lying on the curve \(C_{k}\) for \(k=-47, -39\).