Methods for arriving at numerical solutions for equations of the type [k+3] and [k+5] biquadratics equal to a bi-quadratic for different values of k

TL;DR

This paper introduces methods based on elliptic curve theory to find numerical solutions for specific biquadratic Diophantine equations involving sums of powers equal to quartic forms, addressing a gap in systematic approaches.

Contribution

The authors develop novel numerical solution methods for complex biquadratic Diophantine equations using elliptic curve theory, which was not previously applied systematically.

Findings

Numerical solutions for equations A and B are obtained.

Elliptic curve theory effectively solves these biquadratic equations.

The approach extends the toolkit for solving high-degree Diophantine equations.

Abstract

Different authors have done analysis regarding sums of powers References number 1,2 and 3, but systematic approach for solving Diophantine equations having sums of many biquadratics equal to a quartic has not been done before. In this paper we give methods for finding numerical solutions to equation A given above in section one. Next in section two, we give methods for finding numerical solutions for equation B given above. As is known that finding parametric solutions to biquadratic equations is not easy by conventional method. So the authors have found numerical solutions to equation A and B using elliptic curve theory.