The smallest solutions to the diophantine equation $x^6+y^6=a^6+b^6+c^6+d^6+e^6$

In this paper we discuss a method used to find the smallest nontrivial positive integer solutions to $a_1^6+a_2^6=b_1^6+b_2^6+b_3^6+b_4^6+b_5^6$. The method, which is an improvement over a simple brute force approach, can be applied to search the solution to similar equations involving sixth, eighth and tenth powers.