# “middle” partial denominator in continued fraction expansion of square roots

Suppose $$d$$ is a positive integer that is not a perfect square such that the negative Pell equation, $$x^{2}-dy^{2}=-1$$ has no solution. Then we know the minimal period of the continued fraction expansion of $$\sqrt{d}$$ has even length, $$2\ell$$, and that the partial denominators, $$a_{1},\ldots,a_{2\ell-1}$$, are symmetric about the $$\ell$$-th partial denominator, $$a_{\ell}$$. I.e., $$a_{\ell-j}=a_{\ell+j}$$ for $$j=1,\ldots,\ell-1$$. It is this partial denominator, $$a_{\ell}$$, that I am referring to here as the middle partial denominator.

It is a classical result that $$a_{2\ell}=2a_{0}$$, but my question is what is known about the middle partial denominator, $$a_{\ell}$$, under the above conditions on $$d$$.

Of course, if $$d=a_{0}^{2}+2$$, where $$a_{0}$$ is a positive integer, for example, then we know that the continued fraction expansion of $$\sqrt{d}$$ takes the form $$[a_{0}; \overline{a_{0},2a_{0}}]$$, so the middle partial denominator is $$a_{0}$$ here. But are there more general results known that do not depend on $$d$$ satisfying such quadratic expressions that have nice'' continued fraction expansions?

Any known results with references, ideas,... would be greatly appreciated.

• What you are calling "partial denominator" is usually called "partial quotient". – Gerry Myerson Dec 6 '18 at 11:08
• @GerryMyerson I thought so too, but both wikipedia and wolfram seemed insistent across their continued-fraction-related pages that partial denominator was the term, so I just followed their lead. Happy to change it, especially if it helps get me a good answer here. – user132145 Dec 6 '18 at 18:26