Timeline for What is the simplest, most elementary proof that a particular number is transcendental?
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| Dec 15, 2010 at 1:21 | comment | added | Justin Lanier | Hi, David. Thanks for your answer. Can you expand upon why "With n large enough, nothing arising out of p_(L) can balance these contributions"? I like the approach of focusing on the term of highest power, but I only see that for large n, the contribution is very small, and so I don't see the contradiction. I also like you remark about Cantor and Liouville being equivalent. But I don't see how exactly it could be that each nonzero digit "kills" a collection of polynomials, since if you removed one 1 and left the rest of L the same, wouldn't it still be transcendental? Thanks again. | |
| Dec 14, 2010 at 23:15 | comment | added | Daniel Briggs | It would be interesting to see what portion of Liouville numbers can be taken care of without much trouble by using an appropriate base for expansion and discussing the number in terms of the base (it seems that (1) q must be able to be chosen in a relatively uniform way, such as powers of the base, and (2) the places with the positive digits would have to become sparse enough so that the number wouldn't get muddy: would this requirement be equivalent to the Liouville criterion? Or stronger?) | |
| Dec 14, 2010 at 18:17 | comment | added | Franz Lemmermeyer | This type of idea came up in a letter from Goldbach to Daniel Bernoulli. Goldbach claimed that the number with 1s in every 2^k-th place and 0 everywhere else is irrational because the decimal expansion was not periodic. Liouville, in his article containing his number, actually refers to a letter by Goldbach. | |
| Dec 14, 2010 at 6:46 | comment | added | David Feldman | Hi Daniel Briggs...two minds with but a single thought... your answer appeared just as I posted! | |
| Dec 14, 2010 at 6:43 | history | answered | David Feldman | CC BY-SA 2.5 |