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In this paper, elliptic curves theory is used for solving the Diophantineequations X^3+Y^3+Z^3+aU^k=a_0U_0^{t_0}+...+a_nU_n^{t_n}, k=3,4 where n, ti arenatural numbers and a, a_i are fixed arbitrary rational numbers. We try totransform each case of the above Diophantine equations to a cubic ellipticcurve of positive rank, then get infinitely many integer solutions for eachcase. We also solve these Diophantine equations for some values of n, a, a_i,t_i, and obtain infinitely many solutions for each case, and show among theother things that how sums of four, five, or more cubics can be written as sumsof four, five, or more biquadrates as well as sums of 5th powers, 6th powersand so on.

On the Diophantine equations X^3+Y^3+Z^3+aU^k=a_0U_0^{t_0}+...+a_nU_n^{t_n}, k=3,4