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

Farzali Izadi, Mehdi Baghalagdam

Published 2017-01-06Version 1

The beautiful quartic Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been studied by some mathematicians for many years. Currently, by computer search, the integer solutions of this equation are known for all positive integer values of $h<5000$, and $A, B, C, D<100000$, except some specific 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 several 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 completely solve the Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is an arbitrary rational number. We work out the solutions of the Diophantine equation for some values of $h$, in particular for the values which has not already been found a solution in the range where $A, B, C, D<100000$, by computer search. By our method, we may find infinitely many nontrivial solutions and also obtain infinitely many nontrivial parametric solutions for the above Diophantine equation for arbitrary rational values of $h$.