The 2014 Problem is such a Failure. Let's Try 2015

The mystery theorem is: " Legendre's Three-Square Theorem "

This theorem states that a natural number $n$ can be only expressed as a sum of three squares of integers $x$ , $y$ , and $z$ if and only if $n \neq 4^a (8b + 7)$ where $a$ and $b$ are non negative integers.

Since $2015 = 2008 + 7 = 8(251) + 7$ , therefore $2015$ is expressible as $4^a (8b + 7)$ where $a = 0$ and $b = 251$ .

$\therefore$ the answer is $\boxed{0}$