Q: Foul play in Pell's equation Pells equation $x^2-dy^2=1$   has the obvious x and y solutions $x=n$, $y=1$ for $d=n^2-1$. The next and higher solution for the same d is $x=2n^2-1$ and $y=2n$.
This offers a method for calculation of higher solutions via Brahmagupta's rule (as in Wikipedia under the heading Pell's equation). I then calculate these higher solutions by simply using an excel sheet starting with an entry n and via $d=n^2-1$. I calculate a series of higher solutions for $d=n^2-1$.
I now proceed to do something partly odd or illegal: I simply enter $n=\sqrt n$ instead of $n$. This gives me $d=n-1$, and I now insert $n=\sqrt{62}$ to receive $d=61$ and wondered of I might get integers for both $x$ and $y$ when the "right" numbers came up ($1\, 766\, \,319\, 049$ and $226\, 153\, 980$ for $x$ and $y$ respectively). However in my $5$th solution i.e. my third Brahmagupta iteration I get the numbers $z= 1\, 830\, 972\, 097$ and $y= 234\, 431\, 954.54$.     
Surprisingly these numbers are not far from the real solutions for $d=61$ and they are off both by the same amount, only $3.660$ %.
Could this be just a coincidence?  

A: Yes, the small difference is just a coincidence. If you try the same with some other numbers (such as $d=337$), you'll see find the results of the Brahmagupta iteration starting from non-integer pairs to be considerably different from the "exact" solution.
Of course, the fact that both numbers are off by the same relative amount is trivial since the ratio of $x$ and $y$ must be almost exactly equal to $\sqrt{d}$ (and it gets closer and closer with growing $x$) both for the "exact" and for the "approximate" pairs.
