What are integer solutions to $x^2+7=y^3$

Question

Question: What are the integer solutions to$$x^2+7=y^3\tag1$$

Through Wolfram Alpha, there seems to be only two solutions. Namely, $$\begin{align*} & (x,y)=(1,2)\\ & (x,y)=(181,32)\end{align*}\tag2$$ So I'm wondering about how would you find those solutions. And is there a way to use some sort of transformation to make $(1)$ into a more recognizable form.

And furthermore, is there a formula to determine other type solutions similar to what you already have?

Are you sure about this? Plugging (2,1) in, we get $2^2 +7 = 1^3$. I think you might mean (1,2). — gabe, Dec 08 '16 at 18:37
https://en.wikipedia.org/wiki/Mordell_curve (note that because of the way yours is written, it actually corresponds to $n=-7$ by Wikipedia's notation, not $n=7$.) — Steven Stadnicki, Dec 08 '16 at 18:48
@Frank You can interchange $x$ and $y$ (or rename it), this does not matter. — Dietrich Burde, Dec 08 '16 at 19:32
@Frank Not at all. If $x^2+k=y^3$ then $x^2=y^3-k$; now just swap variables with, e.g., $y'=x$ and $x'=y$ to get $y'^2=x'^3-k$, which is exactly the other given equation. — Steven Stadnicki, Dec 08 '16 at 19:51

marty cohen · Answer 1 · 2016-12-08T20:34:39.133

3

According to this link in the Wikipedia article on the Mordell curve, your solutions are the only ones:

http://tnt.math.se.tmu.ac.jp/simath/MORDELL/MORDELL-

That file claims to have all solutions for $|x^2-y^3| \le 10000 $.

I also recommend this presentation, which was found by a Google search for "mordell equation":

http://www.math.uconn.edu/~kconrad/ross2008/mordell.pdf

edited Dec 08 '16 at 20:34

answered Dec 08 '16 at 20:28

marty cohen

107,799

What are integer solutions to $x^2+7=y^3$

1 Answers1