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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.

Mikolaj
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  • I also don't see any easy argument here. Here is a hard one: https://math.stackexchange.com/a/32903/232 – Qiaochu Yuan Jan 18 '18 at 00:32
  • Is there anything in the original text to suggest that $y>-1$? – Old Peter Jan 18 '18 at 19:11
  • Thank you, Qiaochu; Old Peter, there was no mention of positivity. I think that there was simply a misprint in the original text as no one has seen an obvious elementary method. – Mikolaj Jan 20 '18 at 01:26
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    I still have a nagging doubt that the question might have been correct, and we are missing something obvious. I've searched for $a$ from $-1000$ to $1000$ with $x=0$ to $75000$ and $y=-8255$ to $10$. Indeed, I've found lots of $a$ values that don't show a solution. However, the minimal results such as $46710^2+(-1297)^3=-973$ give me little confidence. – Old Peter Jan 23 '18 at 20:34

1 Answers1

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

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Sam
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    This is completely non-sensical. The question explicitly asks for integer solutions for an arbitrary integer $a$. – Erick Wong Jan 18 '18 at 17:36