A number encoding all primes - MathOverflow

A number encoding all primes

2010-05-11T17:43:35Z

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.

Answer by danseetea for A number encoding all primes

2010-05-11T19:44:38Z

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."

Answer by Matthew Conroy for A number encoding all primes

2010-05-11T21:37:10Z

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.

Answer by Charles for A number encoding all primes

2010-05-12T02:58:55Z

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