# Use of sorting in counterexamples for equations

I came across a question which asked how sorting would help in searching for counterexamples to the conjecture that $$u^6 + v^6 + w^6 + x^6 + y^6 = z^6$$ has no non trivial solutions in integers.

The answer said to make two files containing values of $u^6 + v^6 + w^6 \pmod W$ and $z^6 - y^6 - x^6 \pmod W$. $W$ is the word size of the computer. Sort these, search for duplicates and go for further steps.

Can someone explain what these further steps would be in detail ?

• What are your thoughts on the matter? Have you tried completing this algorithm? – Yuval Filmus Jun 26 '16 at 15:46

You want to find solutions to $u^6 + v^6 + w^6 = z^6 - x^6 - y^6$. Obviously any solution would also have to be a solution modulo k, for every k. So the idea is to find the set of all possible values of $u^6 + v^6 + w^6$ modulo W, and the set of all possible values of $z^6 - x^6 - y^6$, and you need to find the elements that are common to both sets.