From: Marc LeBrun <mlb@well.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, November 13, 2011 3:39 PM Subject: Re: [math-fun] Request for example elementary non-constructive proofs with "witnesses"
Thanks for the many excellent suggestions to think about! Some of them might work; I'll have to ponder further.
Just to be clear, I'm asking for a bit MORE than just an existence proof that gives no hint of how to find an example.
There are indeed a number of notorious cases of those (I appreciate the reminders).
However what I want is one of those, PLUS an actual non-trivial example case that's reasonably easily verifiable.
The example shouldn't be immediately obvious, and would probably be obtained using either more advanced methods, or else just brute force. ... Perhaps Gene's suggestion can be made to work. Certainly the wonderful Cantor diagonal proof is easy enough. But traditionally we'd show the transcendence of some particular example Liouville number X in two stages, by first proving that ALL Liouville numbers are transcendental, and then showing that X is Liouville. Can we keep this argument sufficiently elementary yet make it more direct?
Thanks! _______________________________________________
Mark, Check it out to your own satisfaction. The proof that Liouville numbers are transcendental, and examples of actual Liouville numbers can be found in elementary number theory books, and is surely available on-line somewhere. The proof of the transcendence of e and π can be found in Niven's Carus Monograph, "Irrational Numbers". -- Gene