computability theory - Is the solution bounded Diophantine problem NP-complete?

A particular quadratic Diophantine equation is NP-complete.

$R(a,b,c) Leftrightarrow exists X exists Y :aX^2 + bY - c = 0$

is NP-complete. ($a$, $b$, and $c$ are given in their binary representations. $a$, $b$, $c$, $X$, and $Y$ are positive integers).

Note that there are trivial bounds on the sizes of $X$ and $Y$ in terms of $a$, $b$, and $c$.

Kenneth L. Manders, Leonard M. Adleman: NP-Complete Decision Problems for Quadratic Polynomials. STOC 1976: 23-29