Infinite families of solutions for $A^3 + B^3 = C^3 + D^3$ and $A^4 + B^4 + C^4 + D^4 + E^4 = F^4$

TL;DR

This paper constructs infinite families of solutions for two complex Diophantine equations, extending Ramanujan's identities to generate new solutions for cubic and quartic equations.

Contribution

It introduces novel infinite solution families for Euler's cubic and quartic Diophantine equations inspired by Ramanujan's identities.

Findings

Infinite solutions for $A^3 + B^3 = C^3 + D^3$

Infinite solutions for $A^4 + B^4 + C^4 + D^4 + E^4 = F^4$

Extension of Ramanujan's identities to generate solutions

Abstract

Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler's Diophantine equation $A^3 + B^3 = C^3 + D^3$. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation $A^4 + B^4 + C^4 + D^4 + E^4 = F^4$ in the spirit of Ramanujan.