On the Diophantine equation 𝑥³±𝑦³=𝑎^𝑘±𝑏^𝑘

In this note we consider the title Diophantine equation from both a theoretical as well as experimental point of view. In particular, we prove that for k = 4, 6 k=4, 6 and each choice of the signs our equation has infinitely many coprime positive integer solutions (x, y, a, b) (x, y, a, b) such that no partial sum in the expression x 3 ± y 3 − (a k ± b k) x³ y³- (aᵏ bᵏ) vanishes. The same is true for each k ≢ 0 (mod 4) k 0 4 and the equation x 3 ± y 3 = a k − b k x³ y³=aᵏ-bᵏ. For k = 5, 7 k=5, 7 and all choices of the signs we computed all coprime positive integer solutions (x, y, a, b) (x, y, a, b) of x 3 ± y 3 = a k + b k x³ y³=aᵏ+bᵏ