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

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.