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

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

• $(-1,6)$ and (243,3788) are other solutions May 31, 2016 at 16:59
• $(-1,6)$ is not postive , But Thank you find $(243,3788)$,can you kown this equation have solution? May 31, 2016 at 17:00
• 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 :) May 31, 2016 at 17:03
• Thanks,if you can link the paper May 31, 2016 at 17:04
• 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 May 31, 2016 at 17:16

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 $$=\mathbb{Z}[\sqrt{-k}]$$ 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.