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Find all Mordell's equation:

$$y^2=x^3+k$$ where $k=37$ positive integer numbers,I can't find the when $k=37$ the mordell equation solution with some result,and we can known this equation have solution, such $x=3$ then $3^3+37=64=8^2$ so $(x,y)=(3,8)$ is one solution, can we find other or all positive integer solution, such https://oeis.org/A054504

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  • $\begingroup$ $(-1,6)$ and (243,3788) are other solutions $\endgroup$
    – Bérénice
    May 31, 2016 at 16:59
  • $\begingroup$ $(-1,6)$ is not postive , But Thank you find $(243,3788)$,can you kown this equation have solution? $\endgroup$
    – math110
    May 31, 2016 at 17:00
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    $\begingroup$ Yes I know a theorem to solve Mordell equation with $k$ square =1 or 2 modulo 4 under certain conditions meet here, i will start to write an answer in like 15 min :) $\endgroup$
    – Bérénice
    May 31, 2016 at 17:03
  • $\begingroup$ Thanks,if you can link the paper $\endgroup$
    – math110
    May 31, 2016 at 17:04
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    $\begingroup$ I am back, the link is here math.uga.edu/~pete/4400MordellEquation.pdf I will write an answer in order to not give a link only answer :) Theorem 7 $\endgroup$
    – Bérénice
    May 31, 2016 at 17:16

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You have the following theorem :

Let $k \in \mathbb{Z}^+$ be squarefree, such as $k=1,2 \pmod{4}$, and suppose that the ring $\mathbb{Z}[\sqrt{-k}]$ has the following property : $\forall x,y,z \in \mathbb{Z}[\sqrt{-k}]$, such as $<x,y>=\mathbb{Z}[\sqrt{-k}]$ and $xy=z^3$, $\exists a,b \in \mathbb{Z}[\sqrt{-k}]$ and $\exists$ units $u,v \in \mathbb{Z}[\sqrt{-k}]$ such as $x=ua^n,y=vb^n$. If there exists $a$ such as $k=3a^2\pm1$ then the only solutions to the Mordell equation are $(a^2+k,\pm a(a^2-3k))$.

But you can not apply this theorem directly because the property about $\mathbb{Z}[\sqrt{-37}]$ is not met, but you prove that the property is met on the ring of the algebraic integers in $\mathbb{Z}[\sqrt{-37}]$. See this paper for more details : Paper on Mordell's equation and another one.

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