Finding an integer $c$ for which $a+c^2 \equiv 0 \pmod {2b-2c}$, $a$ & $b$ constant Looking for a solution for such a challenge, I have a decision problem that is solved if there is a positive integer $c$, which for given integer constants $a$, $b$ satisfies the equation
$a+c^2 \equiv 0 \pmod {2b-2c}$
or simply to say $a+c^2$ is divisible by $2b - 2c$.
Additional facts, maybe helpful


*

*$0<c<b$

*$a<2b$


Not sure how to approach this, beside brute force, looking for an elegant solution.

There are many choices for $c$, depending on the given integers $a$ and $b$.
Given 
$a+c^2 \equiv 0 \pmod {2b-2c}$, subtract $b^2$ from both sides;$a+c^2-b^2 \equiv -b^2\pmod {2b-2c}$ 
multiplying through by $2$; 
$2a+(2c-2b)(c+b) \equiv -2b^2 \pmod {2b-2c}$ 
or
$2(a+b^2) \equiv 0 \pmod {2b-2c}$. Or $(a+b^2) \equiv 0 \pmod {b-c}$
From here, input the given values of $a$ and $b$, then equate any factor of $a+b^2$ (that is less than $b$) to $b-c$
