On the Diophantine equation X\(^4\) + hY\(^3\) = Z\(^4\) + hW\(^3\)

Abstract

First we prove that, under a certain condition, the Diophantine equation X⁴ + hY³ = Z⁴ + hW³, where h is a rational number, has infinitely many nontrivial integral solutions.  Using this result we deduce that, under a condition, for k ≥ 2, the Diophantine equation X⁴ + Y\(_{1}^{3}\) + Y\(_{2}^{3}\) + · · · + Y\(_{k}^{3}\) = Z⁴ + W\(_{1}^{3}\) + W\(_{2}^{3}\) + · · · + W\(_{k}^{3}\) has infinitely many nontrivial integral solutions.  Then, we conjecture, by some evidences, that the above conditions may be removed.