Is the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ solvable for any integer $h$?

TL;DR

This paper investigates the solvability of the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ for arbitrary rational $h$, using elliptic curve theory to find new solutions and propose conjectures for general solutions.

Contribution

It introduces a novel elliptic curve approach to find solutions for specific $h$ values and proposes two conjectures for solving the equation for all rational $h$.

Findings

Found new solutions for certain $h$ values not previously known.

Presented parametric solutions for $h$ given by polynomials of degrees 3 and 4.

Proposed two conjectures that could enable solving the equation for any rational $h$.

Abstract

The Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been investigated by some authors. Currently, by computer search, the integer solutions of this equation are known for all positive integer values of $h \le 5000$ and $A, B, C, D \le 100000$, except for some numbers, while a solution of this Diophantine equation is not known for arbitrary positive integer values of $h$. Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees $5$ and $2$ respectively. Also Choudhry presented some new solutions of this equation when $h$ is given by polynomials of degrees $2$, $3$, and $4$. In this paper, by using the elliptic curves theory, we study this Diophantine equation, where $h$ is a fixed arbitrary rational number. We work out some solutions of the Diophantine equation for certain values of $h$, in particular for the values which has not already been found a solution in the range where $A, B, C, D \le 100000$ by computer search. Also we present some new parametric solutions for the Diophantine equation when $h$ is given by polynomials of degrees $3$, $4$. Finally We present two conjectures such that if one of them is correct, then we may solve the above Diophantine equation for arbitrary rational number $h$.