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Abstract. In this paper, the theory of elliptic curves is used for finding the solutions of the quartic Diophantine equation X 4 + Y 4 = 2(U 4 + V 4 ).

In this paper, the elliptic curves theory is used for solving the Diophantine equations a(X ′, where n, m ∈ N∪{0}, and, a, b = 0, a i , b i , are fixed arbitrary rational numbers. We solve the Diophantine equation for some values of n, m, a, b, a i , b i , and obtain nontrivial integer solutions for each case. By our method, we may find infinitely many nontrivial integer solutions for the Diophantine equation for every n, m, a, b, a i , b i , and show among the other things that how sums of some 5th powers can be written as sums of some cubics.

In this paper, we construct a family of elliptic curves with rank ≥ 5. To do this, we use the Heron formula for a triple (A 2 , B 2 , C 2 ) which are not necessarily the three sides of a triangle. It turns out that as parameters of a family of elliptic curves, these three positive integers A, B, and C, along with the extra parameter D satisfy the quartic Diophantine equationKeywords Diophantine equation · elliptic curve · Heron formula 1 IntroductionAs is well-known, the affine part of an elliptic curve E over a field K can be explicitly expressed by the generalized Weierstrass equation of the form E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 ,where a 1 , a 2 , a 3 , a 4 , a 6 ∈ K. In this paper we are interested in the case of K = Q. By the Mordell-Weil theorem [23], every elliptic curve over Q has a commutative group E(Q) which is finitely generated, i.e., E(Q) ∼ = Z r × E(Q) tors , where r is a nonnegative integer called the rank of E(Q) and E(Q) tors is the subgroup of elements of finite order called the torsion subgroup of E(Q).

In this paper we consider a family of elliptic curves of the form y 2 = x 3 −c 2 x +a 2 , where (a, b, c) is a primitive Pythagorean triple. First we show that the rank is positive. Then we construct a subfamily with rank ≥ 4.