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I was embroiled in a small row over this question that I answered regarding integral solutions of $5x^3=y^2+1$. I solved it with elliptic curves and then the OP asked for a solution using the unique factorisation of Gaussian integers, something that had been hinted at in the question body. There was an edit to include this new question, which was then reverted.

Gerry Myerson commented

Are you sure? [at my initial retort of "No" to the query of whether it was solvable with Gaussian integers] The Gaussians are a UFD, $y$ must be even, $\gcd(y+i,y-i)=1$, so $y+i=\alpha(a+bi)^3$ where $\alpha$ is a unit or an associate of $2\pm i$ or of $5$, and so on – occasionally, that approach works.

But I could not flesh out such an approach myself. How can I find all integral solutions to the slightly more general $Nx^3=y^2+1$, where $N$ is a fixed integer, using Gaussian integers?

Parcly Taxel
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    I wrote down $y+i=(2+i)(a+bi)^3$, and it led to $a^3+6a^b-3ab^2-2b^3=1$, and I don't see how to handle this using nothing fancier than the arithmetic of the Gaussian integers. Some Mordell equations $y^2=x^3+k$ can be handled at the level of quadratic fields, but maybe not $5x^3=y^2+1$. – Gerry Myerson Apr 17 '20 at 00:41

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