  In this paper, elliptic curves theory is used for solving the Diophantine
equations X^3+Y^3+Z^3+aU^k=a_0U_0^{t_0}+...+a_nU_n^{t_n}, k=3,4 where n, ti are
natural numbers and a, a_i are fixed arbitrary rational numbers. We try to
transform each case of the above Diophantine equations to a cubic elliptic
curve of positive rank, then get infinitely many integer solutions for each
case. 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 the
other things that how sums of four, five, or more cubics can be written as sums
of four, five, or more biquadrates as well as sums of 5th powers, 6th powers
and so on.
