On a class of quartic Diophantine equations

Abstract

In this paper, by using elliptic curves theory, we study the quartic Diophantine equation (DE) { \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }, where a_i and n\geq3 are fixed arbitrary integers. We try to transform this quartic to a cubic elliptic curve of positive rank. We solve the equation for some values of a_i and n=3,4, and find infinitely many nontrivial solutions for each case in natural numbers, and show among other things, how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for n\geq 3, this paper will be restricted to the examples where n=3,4. Finally, we explain how to solve more general cases (n\geq 4) without giving concrete examples to case n\geq 5.