Find all positive integers that solve Mordell's equation $y^2=x^3+37$

Question

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

Answer 1

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.