# 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

## TL;DR

This paper employs elliptic curve theory to solve complex Diophantine equations involving sums of cubes and higher powers, transforming them into elliptic curves to find infinitely many solutions and explore representations of powers as sums of various powers.

## Contribution

The paper introduces a method to convert specific Diophantine equations into elliptic curves of positive rank, enabling the derivation of infinitely many solutions and representations of powers as sums of other powers.

## Key findings

- Infinite solutions for equations with k=3,4.
- Representation of sums of powers as sums of other powers.
- Application of elliptic curves to solve complex Diophantine equations.

## Abstract

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.