Solve $x^5 + y^5 + z^5 = 2015$ If $x, y, z$ are integer numbers, solve:
$$x^5 + y^5 + z^5 = 2015$$
A friend of mine claims there is no known solution, and, at the same time, there is no proof that there is no solution, but I do not believe him. However, I wasn't able to make much progress disproving his claim.
I tried modular arithmetics, but couldn't reach useful conclusion.

These problems are usually done allowing the variables to have mixed signs, some positive, some negative or zero.
i think I will make this an answer. The similar problem for sums of three cubes has been worked on by many people; as of the linked article, the smallest number for which there are no congruence obstructions but no known expression is $$ x^3 + y^3 + z^3 = 33.  $$
See THIS for the size of numbers involved. Indeed, on the seventh page, they give a list of numbers up to 1000 still in doubt, starts out 33, 42, 74, 156...
I see nothing wrong with suggesting that your problem could be in the same unsettled state, plus I do not think as many people have worked on the sum of three fifth powers.
