On $A^4 + hB^4 = C^4 + hD^4$

TL;DR

This paper proves that the Diophantine equation A^4 + hB^4 = C^4 + hD^4 has solutions for all h less than 20000 and conjectures it holds for all h, contributing to understanding of quartic equations.

Contribution

The author proved solutions exist for h<20000 and conjectured universal solvability for all h, advancing knowledge on this class of quartic Diophantine equations.

Findings

Solutions exist for h<20000

Conjecture: solutions exist for all h

Provides a foundation for future proofs

Abstract

In this paper, this author proved that $A^4 + hB^4 = C^4 + hD^4$ always has the integral solutions for $h < 20000.$ Then we conjecture the equation $A^4 + hB^4 = C^4 + hD^4$ always has the integral solutions.