On the Diophantine equation 𝑥³±𝑦³=𝑎𝑘±𝑏𝑘

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 ) x^3 ± y^3-(a^k ± b^k) 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 x^3± y^3=a^k-b^k . 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 x^3± y^3=a^k+b^k