Timeline for Irrationality of generalized continued fractions
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Apr 11 at 6:56 | history | suggested | Vessel | CC BY-SA 4.0 |
Added variable constraints to ensure continued fractions are indeed infinite.
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| Apr 10 at 16:23 | review | Suggested edits | |||
| S Apr 11 at 6:56 | |||||
| Apr 10 at 16:07 | comment | added | Vessel | Let $a_i=\frac{w_i}{x_i}$ and let $b_i=\frac{y_i}{z_i}$, for $w_i,y_i\in\mathbb{Z}$ and $x_i,z_i\in\mathbb{Z}\setminus\left\{0\right\}$. By applying an equivalence transform, the generalized continued fraction is irrational if $\left|w_{n}x_{n-1}z_{n-1}z_{n}\right|<\left|x_{n}y_{n}\right|$, for all sufficiently large $i$, by Legendre's aforementioned criterion. | |
| Mar 28 at 19:11 | comment | added | Vessel | It should be worth noting that any continued fraction is quadratic irrational iff it's periodic, see: Wolfram MathWorld | |
| Dec 25, 2019 at 12:19 | comment | added | Pazzaz | Do you have a source for Legendre's condition applying when $a_i$ or $b_i$ are negative integers? | |
| Mar 2, 2017 at 1:22 | answer | added | Gottfried Helms | timeline score: 5 | |
| Feb 28, 2017 at 12:47 | answer | added | bhbr | timeline score: 2 | |
| Feb 25, 2017 at 14:17 | comment | added | Gottfried Helms | A better reference is likely David Angell - A family of continued fractions (2010) Journal of Number Theory 130 , pg. 904-911 (Elsevier). | |
| Feb 25, 2017 at 12:49 | comment | added | Gottfried Helms | Perhaps this is interesting: some heuristical examples and systematized at go.helms-net.de/math/divers/GenContFracRationalE.htm | |
| Feb 23, 2017 at 11:15 | comment | added | Gerry Myerson | For the simple continued fraction, the $b_i$ should be positive integers. | |
| Feb 23, 2017 at 9:09 | answer | added | Nemo | timeline score: 8 | |
| Feb 23, 2017 at 8:36 | review | First posts | |||
| Feb 23, 2017 at 8:44 | |||||
| Feb 23, 2017 at 8:23 | history | asked | bhbr | CC BY-SA 3.0 |