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.