Timeline for Constant related to continued fraction of quadratic irrationals
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Aug 7, 2016 at 23:38 | comment | added | Noam D. Elkies | (and if you want the $\liminf$, not the smallest number that works for all $(p,q)$, then the same argument gives $1/(2\sqrt d)$, as Douglas Zare meanwhile wrote as well. The $1 / \sqrt 5$ bound is attained only by irrationals equivlent to the Golden Ratio, which doesn't arise here, not even for $d=5$.) | |
| Aug 7, 2016 at 23:37 | comment | added | Noam D. Elkies | Well $q^2 \left| \frac p q - \sqrt d \right|$ is approximately $|p^2 - d \, q^2| / (2 \sqrt d)$ for large $p,q$, and since we can attain $p^2 - d \, q^2 = \pm 1$ that's the way to go (once we check that even small solutions of $p^2 - d \, q^2 = 2$ can't do better). Units of norm $-1$ do a bit worse than $1 / (2 \sqrt d)$, and units of norm $+1$ do a bit better, the difference decreasing with $p$ and $q$. So the smallest $+1$ solution yields the best $c_d$. | |
| Aug 7, 2016 at 22:29 | comment | added | Stanley Yao Xiao | Can you elaborate on the statement "Seems that the optimal $c_d$ comes from the minimal positive solution of $p^2 - dq^2 = +1$"? This seems to be close to exactly what I want | |
| Aug 7, 2016 at 22:16 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |