[math-fun] Gelfond-Schneider, Lindemann-Weierstrass, and formatting
Eugene Salamin <gene_salamin@yahoo.com> wrote:
Yahoo mail messed up the formatting, so here is a corrected version.
Gelfond?Schneider theorem:? If a and b are algebraic numbers with a???0,1 and if b is not a rational number, then any value of a^b is a transcendental number.
The digest option turns all non-ASCII characters into question marks. As such, I recommend avoiding non-ASCII characters on this list. As you say, the Gelfond-Schneider theorem involves raising an algebraic number to an algebraic power. But we were discussing raising a transcendental number (e) to an algebraic power, so Lindemann-Weierstrass is the relevant theorem. No, wait. We were discussing raising a transcendental number (e) to a transcendental power (pi*sqrt(163)). So I don't know what theorem applies, if any. Maybe e^(pi*sqrt(n)) really can be an integer for some nonzero positive integer n? It is an integer for n = -1, after all. Dan Asimov <dasimov@earthlink.net> wrote:
P.S. This reminder by Keith led me to review that April 1975 column (which I have on my computer from a CD I got from the MAA). Fun!
I too reviewed it, mostly to make sure I was remembering the year correctly. (I was.) But I still have the original magazine. I'm such a packra^H^H^H archivist.
Well, e^(pi*i) = -1 is algebraic not in {0,1}, and -i*sqrt(163) is an irrational algebraic. So Gelfond-Schneider applies. —Dan On Mar 2, 2014, at 8:52 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
We were discussing raising a transcendental number (e) to a transcendental power (pi*sqrt(163)). So I don't know what theorem applies, if any. Maybe e^(pi*sqrt(n)) really can be an integer for some nonzero positive integer n? It is an integer for n = -1, after all.
On Sun, 2 Mar 2014, Keith F. Lynch wrote:
Eugene Salamin <gene_salamin@yahoo.com> wrote:
Yahoo mail messed up the formatting, so here is a corrected version.
Gelfond?Schneider theorem:? If a and b are algebraic numbers with a???0,1 and if b is not a rational number, then any value of a^b is a transcendental number.
The digest option turns all non-ASCII characters into question marks. As such, I recommend avoiding non-ASCII characters on this list.
I don't think that's the problem. I'm pretty sure that the digest is encoded in UTF-8 (or can be -- it might be a mailman option.) Especially for math stuff, Unicode is a big improvement over ASCII. Restricting to ASCII would be fairly unpleasant. I read the digest and Gene's zetas came through just fine. I usually use Pine (actually Alpine Vers. 2.11) on Linux to read mail. I had to tweak a few things to get it right, but once everyone (mailman, pine, terminal emulator) agreed that UTF-8 was the appropriate text representation, everything just started working. I also read the math-fun digest on my iPhone (the opposite end of the email reading technology curve) occasionally, and that mostly works fine too. -- Tom Duff. If I could speak English, I would lecture the loud foreigners on the train.
participants (3)
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Dan Asimov -
Keith F. Lynch -
Tom Duff