This question began with Dror Speiser's answer to
Dror has some ideas on how to proceed. Here is what I had in mind when I mentioned it. This is a really open-ended problem, but should not be thought of as an open problem yet because I just made it up.
What I have in mind is the definition of some measurement of shortness $S(n)$ for the period length of the continued fraction for $\sqrt n$ where $n$ is a positive integer. Then I am hoping an optimization problem and for a sequence of numbers $n_t$ in the style of the colossaly abundant numbers, whereby $S(n_t)$ would automatically be better than $S(k)$ for any $k < n.$ The payoff could be explicit factorizations for the $n_t.$
A very nice paper on this is J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988. See also
http://en.wikipedia.org/wiki/Superabundant\_number
http://en.wikipedia.org/wiki/Colossally\_abundant\_number
http://en.wikipedia.org/wiki/Highly\_composite\_number
http://en.wikipedia.org/wiki/Superior\_highly\_composite\_number
Let me give some detail.
(I) The first step, and I am not utterly certain this is known, is do we even know for sure that the period lengths increase without bound? This would appear to be explicit in Golubeva (1991),
http://www.springerlink.com/content/f11602324177865w/
(II) Next we need some explicit increasing lower bound $f(n).$ Let us define $l(n)$ as the period length of the continued fraction for $\sqrt n$ where $n$ is a positive integer. Again, I do not know whether such a bound $f(n) \leq l(n)$ is known.
(III) Third is some tiny (but increasing) function $g(n)$ that grows so slowly that for any real $t > 1,$ $$ \lim_{n \rightarrow \infty} \frac{(g(n))^t}{l(n)} = 0. $$ If it turns out to be suitable, we could just take $g(n) = \log f(n).$
(IV) It follows that $$ \frac{(g(n))^t}{l(n)} $$ achieves a maximum. If the maximum is achieved by only one value of $n,$ then that is the definition of $n_t.$ If the maximum is achieved at more than one value, take $n_t$ to be the largest of these (at least that is what is done with the superior highly composite numbers, although the point is not made clear in the Nicolas article).
(V) If I have got the order correct on the inequalities, we would hope to get a sequence $n_t$ that is constant on intervals $t_1 < t < t_2$ but eventually increases. The jump points at which $n_t$ increases would be of interest. With any luck, some reasonable measure of surprising shortness would be improving as the $n_t$ increase. Finally, the most optimistic part of this is the possibility that, as in Nicolas and Alaoglu and Erdos before that and Ramanujan before that, there is an explicit factorization of $n_t.$
Well, thanks for reading this far.
