Consider the negative Pell equation: $x^2-10y^2=-1$
Its integer positive fundamental solution is $(x_1, y_1)=(3,1)$. On the online solver (https://www.alpertron.com.ar/QUAD.HTM), we know that:
$x_{n+1}=19x_n+60y_n$
$y_{n+1}=6x_n+19y_n$
By applying those formulas, we obtain the following consecutive solutions to the Pell equation: $(117,37), (4443,1405), (168717,53353), (6406803,2026009)$.
Next solution should be: $(243289797, 76934989)$, but using these values, we get:
$243289797^2-10(76934989^2)=0$
Consequently also the next solution, produce the same result:
$9238605483^2-10(2921503573^2)=0$
Why the recurrence formula generated by the online solver (but also by a solver I'm programming: same results) give this error for bigger solution?
You are using limited-precision numbers, I'd guess.
Since $243289797^2$ is odd and $10(76934989^2)$ is even, their difference can't be $0$.
Computing with arbitrary precision, I get $-1$ for both values.