TITLE: Efficient computation of the least fraction with square denominator greater than the square root of 2.
QUESTION [8 upvotes]: The least rational number greater than $\sqrt{2}$ that can be written as a ratio of integers $x/y$ with $y\le10^{100}$ can be found in a moment using a little Python program. Can anyone write a program that finds, in hours rather than centuries, the least rational greater than $\sqrt{2}$ of the form $x/y^2$ with $y^2\le 10^{100}$? 
More generally, my question is whether the following computation is known to be feasible or not feasible:
Given $N$, find  the least rational greater than $\sqrt{2}$ of the form $x/y^2$,
 with $x$ and $y$ integers and $y^2\le N$. For definiteness, let's say that the output should be the required rational written in lowest form.
By a feasible computation I mean one that can be done in $O((\log N)^k)$ bit operations for some constant $k$. 
Of course the square root of 2 is not essential here. Any irrational would do, as long as comparisons with rationals are feasible. I don't know of any such irrational for which I can answer the question I've posed.

REPLY [2 votes]: You should try the algorithms in Elkies' paper (from 2000) "Rational points near curves ..." http://arxiv.org/abs/math/0005139 .  His idea is to cover the curve with a bunch of small rectangles, and use lattice basis reduction within each such region.  He proves a result which either says that there are small number of solutions or all the solutions lie on a line.