A Note on the Quartic Diophantine Equation $A^4+hB^4=C^4+hD^4$

TL;DR

This paper explores solutions to the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ for various polynomial values of $h$, extending known solutions to new polynomial degrees.

Contribution

It introduces new solutions for the equation when $h$ is a polynomial of degrees 2, 3, and 4, expanding the known solution space beyond previous polynomial degrees.

Findings

New solutions for $h$ as quadratic, cubic, and quartic polynomials.

Extension of known solutions beyond $h<1000$.

Demonstrates the existence of solutions for broader polynomial classes.

Abstract

Integer solutions of the diophantine equation $A^4+hB^4=C^4+hD^4$ are known for all positive integer values of $h < 1000$. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of $h$ is not known, Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees 5 and 2 respectively. In this paper, we present several new solutions of this equation when $h$ is given by polynomials of degrees $2,\;3$ and 4.