https://artofproblemsolving.com/community/q1h1473536p8562327

RagvaloD wrote: https://artofproblemsolving.com/community/q1h1473536p8562327 Almost; this one asks for the smallest non-special number.

$1$ is special, proved in link $2$ is special, because: $(2^m(n+1))^2+(2^mn)^2-2(2^mn)^2=2^{2m}(2n+1)$ $(2^m(n+1))^2+(2^m(n+1))^2-2(2^mn)^2=2^{2m+1}(2n+1)$ $3$ is not special, because: $a^2+b^2-3c^2=3$ has no solutions. Proof: Let such $a,b,c$ exists, then $a=b=0 \pmod 3 \to a=3a_1,b=3b_1$ $3a_1^2+3b_1^2-c^2=1$ but $3a_1^2+3b_1^2-c^2=0,2 \pmod 3$ -contradiction

To prove that for every integer $n$ there exist integers $a, b$ and $c$ such that $a^2+b^2-2017c^2=n$, let us consider two cases: • Even $n$: take $(a,b,c)=(44\cdot (\frac{n}{2}-13)-1,9\cdot (\frac{n}{2}-13)+5,\frac{n}{2}-13)$. • Odd $n$: take $(a,b,c)=(44\cdot \frac{n-1585}{2}+8, 9\cdot \frac{n-1585}{2}-39, \frac{n-1585}{2})$.

Motivation for solution, anyone?