Algorithm for diophantine equation From p. 38 of Richard Hodel's An Introduction to Mathematical Logic,

"Construct an algorithm that, given an integer $a$, decides in a finite number of steps whether the equation $x^2+y^3-a=0$ has a solution in integers."

I don't see how to place any bound on $x$ or $y$ in terms of $a$, and given the nature of the text I don't think Hodel was expecting any heavy-duty number theory.

POSTSCRIPT: It seems most likely that Hodel meant "a solution in POSITIVE integers", as I don't think the author was expecting familiarity with the  bound mentioned by Qiaochu.
The above equation is shown below:

$$x^2+y^3-a=0  \tag1\label1$$

Consider restricting value of $a$ to $a=(x+y)^2$ then, the solution to equation \eqref{1} above is given below:

\begin{align}
x &amp;= \frac{6k^2-7k+2}{k^2} \\
y &amp;= \frac{-3k^2+2k}{k^2}.
\end{align}
The above equation is shown below:

$$x^2+y^3-a=0  \tag1\label1$$

Consider restricting value of $a$ to $a=(x+y)^2$ then, the solution to equation \eqref{1} above is given below:

\begin{align}
x &amp;= \frac{6k^2-7k+2}{k^2} \\
y &amp;= \frac{-3k^2+2k}{k^2}.
\end{align}