Timeline for Patterns in Generalized Continued Fractions
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 18, 2020 at 18:50 | review | Suggested edits | |||
| Aug 18, 2020 at 19:41 | |||||
| Jun 25, 2013 at 3:02 | review | Late answers | |||
| Jun 26, 2013 at 16:16 | |||||
| Jan 15, 2012 at 14:56 | comment | added | Douglas Zare | I'm well aware of the theory of simple continued fractions. Your statement that finding a GENERALIZED continued fraction for $\gamma$ would settle the rationality of $\gamma$ appears wrong. | |
| Jan 13, 2012 at 3:23 | comment | added | Glenn | For rationality, if the SIMPLE (!) continued fraction for an expression terminates after a finite number of terms, the expression is rational; otherwise, it is not. Your above example shows that rational numbers can have nonterminating GENERALIZED CFs, but that's because there can be infinitely many of those for a given number, as opposed to only one SIMPLE CF (two if the final term b(k) > 1 is re-expressed as [..., b(k)-1, 1], but this is trivial). I try to choose my CFs to combine speed with relative simplicity. | |
| Dec 17, 2011 at 19:37 | comment | added | Douglas Zare | I still see no universal test for rationality. | |
| Nov 27, 2011 at 4:07 | comment | added | Glenn | If you look more closely in the "Examples" section, you'll see that your GCF for "2" is a special case of the GCF for the square root of (x^2+y), where in your case x=1 and y=3. It's certainly possible for nth roots be rational, but they generally have a simpler form than transcental functions and numbers. I expect that should a pattern be found, it will be cyclic, but the numbers themselves will not repeat, similar to the nth root examples in the same article, but requiring several terms before the repeating pattern becames apparent. | |
| Oct 26, 2011 at 10:28 | comment | added | Douglas Zare | How would finding a generalized continued fraction determine whether $\gamma$ is irrational? $2=1+3/(2+3/(2+3/(2+...$. | |
| Oct 26, 2011 at 1:14 | history | answered | Glenn | CC BY-SA 3.0 |