Timeline for Is pi = log_a(b) for some integers a, b > 1?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 12, 2013 at 16:32 | vote | accept | Stefan Kohl♦ | ||
| Jan 11, 2013 at 17:49 | answer | added | Timothy Chow | timeline score: 33 | |
| Jan 10, 2013 at 14:41 | comment | added | Stefan Kohl♦ | @Emil: Indeed Nesterenko proved 'only' algebraic independence of $\pi$, $e^\pi$ and $\Gamma(\frac{1}{4})$. -- Thank you very much for pointing this out! Also, interesting that transcendence or even only irrationality of $e+\pi$ and $e\pi$ still has not been settled. | |
| Jan 10, 2013 at 14:36 | comment | added | Stefan Kohl♦ | @Ricky: Obviously one can ask the question for other transcendental numbers as well - but I think I am maybe not the only one who regards π as a number of particular interest. So far I don't know examples of transcendental numbers x not defined in terms of logarithms such that ax=b for some integers a,b>1. | |
| Jan 10, 2013 at 12:43 | comment | added | Emil Jeřábek | Concerning the linked page: as far as I know, $e$ and $\pi$ are not known to be algebraically independent, and in fact, $e+\pi$ and $e\pi$ are still not known to be irrational. What Nesterenko proved is that $\pi$, $\color{red}{e^\pi}$, and $\Gamma(1/4)$ are algebraically independent. | |
| Jan 10, 2013 at 12:10 | comment | added | Ricky | Is there a particular reason to consider $\pi$ and not other transcendental numbers? For example, is the problem known for $e$ or $\sum_i 10^{-i!}$? Just for curiosity, can you give an example of a number $x$ such that $a^x=b$ but $x$ is not defined as $\log_a(b)$ (I know this not a precise question)? | |
| Jan 10, 2013 at 11:37 | history | asked | Stefan Kohl♦ | CC BY-SA 3.0 |