I usually work in the field of differential geometry, but I have encountered the following problem in my research: Are there infinitely many positive integers $k,l,m\in\mathbb N^{>0}$ such that $$(3+3k+l)^2=m\,(k\,l-k^3-1)\,?$$ Obviously, taking $l=k^2$
and $m=-(3+3k+l)^2$
gives infinitely many integer solutions, but $m<0$ is negative. As a non-expert, I imagine that there is either a simple answer to this question, or the problem is not so simple to solve. Of course, I've played around with the equations a bit, but other than finding numerous examples, I haven't made any progress.


I would appreciate an existence or non-existence statement for infinitely many positive integer solutions, but also some hints that the problem is most likely hard to solve would help me.


Background: I am looking for certain integer representations of a surface group, and I can show that integer solutions to this diophantine equation actually give rise to integer representations. The condition that $k,l,m$ are positive is equivalent to the condition that the corresponding representation is contained in a higher Teichmüller component (which is important for my differential geometric application).



Are there infinitely many positive integer solutions to $(3+3k+l)^2=m\,(k\,l-k^3-1)$?
It does have infinitely many positive solutions. Here is just one such series.


Consider the following recurrence sequence:


$$u\_0=1,\ u\_1=2,\ u\_{n+1} = 23 u\_n - u\_{n-1} - 4\qquad (n\geq 1).$$


Let $t,k$ be any two consecutive terms of this sequence, then setting $l:=k^2+t$ produces the following equality:
$$(3+3k+l)(t+1) = (k+26)(kl-k^3-1),$$
which gives solution $m:=\frac{(k+26)(3+3k+l)}{t+1}$ (which is an integer) to the original equation.




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In fact, integrality of $m$ follows from the identity:
$$(u\_{n+2}+1)(u\_n+1) = (u\_{n+1}+26)(u\_{n+1}+1),$$
which can be verified from the recurrence for $u\_n$.


In summary, the values $(k,l,m)$ in this solution series are given by
$$\begin{cases}
k = u\_{n+1}, \\
l = u\_{n+1}^2 + u\_n, \\
m = (u\_{n+2}+2)(u\_{n+1}+2) + 24.
\end{cases}\qquad (n\in\mathbb{Z}\_+)
$$




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**ADDED.** I've added $u\_n$ to the OEIS as [sequence A350917](https://oeis.org/A350917). Together with 9 other similar recurrences it gives all solutions $k$ to $(tk-1)\mid (k+1)^4$, which are now listed in [sequence A350916](https://oeis.org/A350916).


