EQUAL SUMS OF LIKE POWERS

AJAI CHOUDHRY

Abstract:

This paper is concerned with four diophantine systems, namely, (i)  $\sum^4_{i=1}x^k_i=\sum^4_{i=1}y^k_i$, $k=1,2,4$; (ii)  $\sum^4_{i=1}x^k_i=\sum^4_{i=1}y^k_i$, $k=1,3,4$; (iii)  $\sum^4_{i=1}x^k_i=\sum^4_{i=1}y^k_i$, $k=2,3,4$; (iv)  $\sum^3_{i=1}x^k_i=\sum^3_{i=1}y^k_i$, $k=2,3,4$. Parametric solutions as well as numerical examples of solutions in positive integers of the first three diophantine systems have been obtained in the paper. For the fourth diophantine system, solutions do not exist in positive real numbers and a single numerical solution in integers has been obtained.