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Abstract

In this paper, elliptic curves theory is used for solving the quartic Diophantine equation X 4 + Y 4 = 2U 4 + n i=1 T i U 4 i , where n ≥ 1, and T i , are rational numbers. We try to transform this quartic to a cubic elliptic curve of positive rank, then get infinitely many integer solutions for the aforementioned Diophantine equation. We solve the above Diophantine equation for some values of n, T i , and obtain infinitely many nontrivial integer solutions for each case. We show among the other things that some numbers can be written as sums of some biquadrates in two different ways with different coefficients.