Solutions to equation (or proof that there is no solution) I am unable to find a solution or prove that there is none. I really need help:
$a, b,$ and $c$ are positive integers such that
$a^2-b^2+c^2=2.$
Is this possible? If possible find the solutions, if not prove that there is no solutions.

Rewrite the equation as $c^2-2=b^2-a^2$. If $c$ is odd, then so are $c^2$ and $c^2-2$. Every odd number is the difference between two consecutive squares. That gives us an infinite family of solutions: For every $k=1,2,\ldots$, let
$(a,b,c)=(2k^2+2k-1, 2k^2+2k,2k+1)$
For $k=1,2,3,4,5$, this gives us solutions:
$(3,4,3)\\
(11,12,5)\\
(23,24,7)\\
(39,40,9)\\
(59,60,11)$
These aren't all the solutions, but it's one infinite family. Indeed, one example not fitting this pattern is $(17,24,17)$
