Timeline for What is the set of all "pseudo-rational" numbers (see details)?
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19 events
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| Jan 17, 2017 at 8:21 | comment | added | Wojowu | @ASKASK But they are closed under multiplication by rational numbers: if $e/\pi=k/l$ and $\pi=\sum P(n)/Q(n)$ (see e.g. Matt F.'s answer below), then $e=\sum kP(n)/lQ(n)$. | |
| Jan 17, 2017 at 5:57 | comment | added | ASKASK | @RobertIsrael Are you saying that a proof that $e$ is not psuedo-rational would imply that $e/\pi$ is not rational? I'm not sure I follow, since this set doesn't seem to be closed under division | |
| Jan 17, 2017 at 1:34 | comment | added | Robert Israel | @PerAlexandersson Guesses are easy to make, proofs are something else. | |
| Jan 17, 2017 at 1:33 | comment | added | Robert Israel | It's still an open question whether $e/\pi$ is irrational, so don't expect a proof that $e$ is not pseudorational. | |
| Jan 17, 2017 at 1:28 | answer | added | Robert Israel | timeline score: 7 | |
| Jan 16, 2017 at 15:42 | answer | added | Gerald Edgar | timeline score: 5 | |
| Jan 16, 2017 at 14:14 | comment | added | Per Alexandersson | I would guess that Liouvilles constant, mathworld.wolfram.com/LiouvillesConstant.html is probably not pseudo-rational... | |
| Jan 16, 2017 at 4:37 | answer | added | user44143 | timeline score: 11 | |
| Jan 15, 2017 at 12:12 | comment | added | user44143 | @GerhardPaseman, calculating to within epsilon is routine, but that procedure won't tell you about positivity. How would it work for $P/Q = 6/n^2 - 8/(2n-1)^2$ ? | |
| Jan 15, 2017 at 6:20 | comment | added | Gerhard Paseman | Since the terms either diverge or grow like $O(n^{-2})$, I think determining positivity and good bounds for the value are routine, especially for small degree denominators. Gerhard "In Theory If Not Practice" Paseman, 2017.01.14. | |
| Jan 14, 2017 at 20:11 | comment | added | user44143 | Given P,Q, can we decide whether the pseudorational is positive, or greater than a given algebraic number? How many terms would we need to calculate? Those are easier than @KevinBuzzard's question, but still not trivial. | |
| Jan 14, 2017 at 11:14 | comment | added | Kevin Buzzard | This question might be completely inaccessible. for example $\zeta(7)$ is pseudorational and hence any rational multiple of it is too. But it could maybe be an open problem to even say whether $e$ is a rational multiple of $\zeta(7)$ -- these things are hard. Maybe someone can come up with a clever proof that (some well-known number) is pseudorational, but most numbers won't be and my guess is that you won't be seeing anyone posting proofs that (some well-known number) is not. Here's a question which might be more tractible -- are pseudorational numbers all periods in the sense of Zagier? | |
| Jan 14, 2017 at 7:43 | comment | added | Gerhard Paseman | Also, you should place certain restrictions on Q and P to avoid division by zero and diverging sums. It includes values of the zeta function in addition to pi^2/6, and likely some more exotic numbers. In addition to checking for algebraic and Liouville numbers, I suggest looking at some mathematical constants that may arise from this, particularly work of Steven Finch. Also, approximation (by rational numbers) theory may help. Gerhard "Try Diverging Away From Nullity" Paseman, 2017.01.13. | |
| Jan 14, 2017 at 7:30 | comment | added | Seva | Expanding on Gerhard Paseman's comment, the set in question is in fact a subspace of the reals considered a vectr space over the rationals | |
| Jan 14, 2017 at 7:30 | history | edited | Andrew Lin |
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| Jan 14, 2017 at 7:17 | comment | added | Gerhard Paseman | It is an additive subgroup of the reals in which each number is "slowly"approximable. I don't know if all algebraic numbers are in this group. I am pretty sure it is not closed under multiplication and that e is not in this subgroup, but I can't prove it yet. Gerhard "Perhaps It Excludes Liouville Numbers?" Paseman, 2017.01.13. | |
| Jan 14, 2017 at 7:05 | history | edited | Andrew Lin | CC BY-SA 3.0 |
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| Jan 14, 2017 at 6:56 | review | First posts | |||
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| Jan 14, 2017 at 6:51 | history | asked | Andrew Lin | CC BY-SA 3.0 |