Nice. I had a simpler example in mind, namely one that follows from *only* the following facts about algebraic numbers: -- The algebraic numbers form a field of characteristic zero; -- x and exp(x) cannot both be algebraic. I believe the following extension puzzle has no such elementary solution: "Find x such that x, exp(x) and exp(exp(x)) are all transcendental." Sincerely, Adam P. Goucher
Sent: Wednesday, August 12, 2015 at 5:19 AM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] A transcendental puzzle
Well, I do know the answer, as it happens --- so does APG, no doubt.
Anybody determined to cheat (not that anybody would, of course) can always consult https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem
WFL
On 8/12/15, Keith F. Lynch <kfl@keithlynch.net> wrote:
Fred Lunnon <fred.lunnon@gmail.com> wrote:
However, I did slip up again: gamma has not actually been proven transcendental! I should have employed exp(1), ...
Indeed, exp(x) is transcendental for all non-zero algebraic x. It follows from that that if exp(x) is algebraic, x must be transcendental. Hence, for instance, the natural log of 2 must be transcendental.
But since the transcendental numbers are of a higher cardinality than the algebraic numbers, it must be that in "most" cases x and exp(x) are both transcendental.
Puzzle: Give an example where x and exp(x) are both known to be transcendental. I'll give a solution in a week if nobody posts one sooner.
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