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In this paper we discuss a method used to find the smallest non-trivial positive integer solutions to a16 + a26 = b16 + b26 + b36 + b46 + b56. 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.

The smallest solutions to the diophantine equation 𝑥⁶+𝑦⁶=𝑎⁶+𝑏⁶+𝑐⁶+𝑑⁶+𝑒⁶