I am trying to get a recurring solutions for this kind of Diophantine equation: $XY=a^2(ab+1)=K$; where $a,b \in$ $\Bbb N$ and $a=even$
First of all, I am industrial engineer, sorry for that :) and if I am trying to do something Math. impossible
I tried it with Alpertron method but I become an identity type like $X_{n+1}=X_n$ https://www.alpertron.com.ar/METHODS.HTM#SolGral
I also had a look to this Question but I could not guess help for me Proof that $x^2 - 2y^2 = -1$ has a recurring solution for $x$
QUESTION: can somebody help me with a clue to get a recurring solutions without finding all divisors of K? $\implies$ I suppose that it means not to make calculations to find $(ab+1)$ divisor ones
Ex: for $a=6$ and $b=8$, K=1764. There are 27 $\in\Bbb N$ solutions ¿are the 26+1 predictable with a general recurring formula, just knowing in advance 1 solution of them?