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.

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.
