Welcome to MathOverflow

Question

9

2

This may be a soft question, but it's just something I thought of one night before sleeping. It's not my field at all, so I am just asking out of curiosity. Has anyone studied the number which is the sum over primes $\sum{ 2^{-p}}$? Its binary expansion (clearly) has a 1 in each prime^th "decimal place", and a zero everywhere else, so, it should be important in number theory I would guess.

flag	asked May 11 2010 at 17:43 David Carchedi 7,707●10●27

5

No, unlikely to be of interest in number theory. – Gerald Edgar May 11 2010 at 18:20

9

Why not use $\sum 10^{-p}$ instead, so you can remove the quotes around decimal place? Anyway, numbers of this form are nothing else than a curiosity but of little use. Even though they encode information about all primes, you need to input all the same information in the definition of the number. – Álvaro Lozano-Robledo May 11 2010 at 18:20

@Alvaro: That's fair enough. I used 2 because what made me think of this was an old homework problem from a number theory course I took in Russia: Write a "formula" for the n^th prime. You can use a similar "trick" as above to do so. – David Carchedi May 11 2010 at 18:39

3

Compare with the product formula for the Riemann zeta function: tinyurl.com/29bythb With the zeta function you elegantly get prime numbers combining together in all possible ways to form all natural numbers raised to the power of $s$. The whole thing is very natural. Your sum doesn't really have such properties. If you start trying to form powers of it (say) you get all kinds of "cross" terms that make it hard to assign meaning to the expansion. – Dan Piponi May 11 2010 at 19:55

1

there are numbers that also encode this thread; check them: they also include a lot of variations that we are ashamed to try here ;) – Pietro Majer Oct 28 at 21:16

danseetea · Accepted Answer · 2010-05-11 19:44:38Z UTC

Here is Hardy & Wright's answer from "An Introduction to the Theory of Numbers", (5th ed, p344), where they discuss a similar number:

"Although ... gives a 'formula' for the nth prime, it is not a very useful one. To calculate $p_n$ from this formula, it is necessary to know the value of $a$ correct to $2^n$ decimal places; and to do this, it is necessary to know the values of $p_1$, $p_2$, ..., $p_n$ ... There are a number of similar formulae which suffer from the same defect ... Any one of these formulae (or any similar one) would attain a different status if the exact value of the number $a$ which occurs in it could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible."

Matthew Conroy · Answer 2 · 2010-05-11 22:04:55Z UTC

8

You might take a look at the paper by Forenc Adorjan, "Binary Mappings of monotonic sequences and the Aronson function". It specifically discusses the number you describe.

edited May 11 2010 at 22:04

answered May 11 2010 at 21:37

Matthew Conroy
211●1●4

Charles · Answer 3 · 2011-09-03 23:13:49Z UTC

6

See http://oeis.org/A051006 and http://mathworld.wolfram.com/PrimeConstant.html which cover this particular sequence.

edited Sep 3 2011 at 23:13

answered May 12 2010 at 2:58

Charles
4,139●8●26

A number encoding all primes

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.