Prove or disprove transcendence of numbers

Question

I have two in one question:

1) Let $\{p_n\}_{n\in \mathbb{N}}$ be sequence of all prime numbers. Is number $\displaystyle\alpha = \sum_{n=1}^{\infty} 10^{-p_n}$ transcendental number?

2) Let $\{F_n\}_{n \in \mathbb{N}}$ be Fibonacci sequence with initial values $F_1=1$ and $F_2=2$. Is number $\displaystyle\beta = \sum_{n=1}^{\infty} 10^{-F_n}$ transcendental number?

If anyone thinks of something better to retag it with, be my guest.

Usually people here prefer if you ask one question per post, to avoid this annoying situation in which someone can answer one of the questions and not the other. — MJD, Feb 24 '13 at 17:53
For the first question see here and this MathOverflow question. The question seems to be open. — Martin, Feb 24 '13 at 18:02

André Nicolas · Accepted Answer · 2013-02-24T19:18:23.713

7

Your second number has been proved transcendental. A somewhat more general result about numbers of the form $\sum 10^{-\lfloor \beta^k\rfloor}$ is mentioned in the Wikipedia list of numbers that have been proved transcendental.

Added: It has been pointed out that the Wikipedia reference is, to put it mildly, thin. To chase down information about Sturmian sequences, one can for example look at this article and its references. I believe that the original result is due to Mahler.

edited Feb 24 '13 at 19:18

answered Feb 24 '13 at 17:59

André Nicolas

514,336

is that it? mentioned with no citation on wikipedia? – Feb 24 '13 at 18:55

Prove or disprove transcendence of numbers

1 Answers1