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.
 A: 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.
A: Since the question doesn't mention signs, then equivalently,
$$x^n+y^n = z^n+\beta$$
This is what Noam Elkies calls a Fermat near-miss and he has a table for $n\leq20$. It is interesting to ask: Let $xyz\neq0$. For a given $n$, how small can $\beta$ get?

For $n=3$: 

It is well-known that $\beta =1$. (And it has an infinite number of integer solutions.)

For $n=5$: 

It seems it is $\beta = 12$,
$$13^5+16^5=17^5+12$$
D. Stork's search showed there is no $\beta =2015$ with $|x|,|y|,|z|\leq200$. We can extend that with Elkies' tables (which go as high as $8\; million$). Excluding $|x|,|y|,|z|<17$, the next smallest $|\beta|$ with $\gcd(x,y,z)=1$ are, 
$$\begin{aligned}
&42^5 + 71^5 = 72^5 + 2951\\
&104^5 + 133^5 = 140^5-75083\\
&133^5 + 228^5 = 231^5-87890\\
&\quad\vdots\\
&707902^5 + 5645541^5 = 5645576^5+39515947850357
\end{aligned}$$
So $\beta=2015$ has missed the train, and the chances it exists is very remote.
A: A very quick exhaustive computer search (for $0 \le x, y, z \leq \lceil 2015^.2 \rceil = 5$) shows there is no solution, as does a search with $0 \le x \le 200$ and $-200 \le y, z, \le 200$.
