A number encoding all primes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:44:19Z http://mathoverflow.net/feeds/question/24270 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24270/a-number-encoding-all-primes A number encoding all primes David Carchedi 2010-05-11T17:43:35Z 2011-09-03T23:13:49Z <p>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.</p> http://mathoverflow.net/questions/24270/a-number-encoding-all-primes/24283#24283 Answer by danseetea for A number encoding all primes danseetea 2010-05-11T19:44:38Z 2010-05-11T19:44:38Z <p>Here is Hardy &amp; Wright's answer from "An Introduction to the Theory of Numbers", (5th ed, p344), where they discuss a similar number:</p> <blockquote> <p>"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."</p> </blockquote> http://mathoverflow.net/questions/24270/a-number-encoding-all-primes/24296#24296 Answer by Matthew Conroy for A number encoding all primes Matthew Conroy 2010-05-11T21:37:10Z 2010-05-11T22:04:55Z <p>You might take a look at <a href="http://web.axelero.hu/fadorjan/aronsf.pdf" rel="nofollow">the paper by Forenc Adorjan, "Binary Mappings of monotonic sequences and the Aronson function"</a>. It specifically discusses the number you describe.</p> http://mathoverflow.net/questions/24270/a-number-encoding-all-primes/24324#24324 Answer by Charles for A number encoding all primes Charles 2010-05-12T02:58:55Z 2011-09-03T23:13:49Z <p>See <a href="http://oeis.org/A051006" rel="nofollow">http://oeis.org/A051006</a> and <a href="http://mathworld.wolfram.com/PrimeConstant.html" rel="nofollow">http://mathworld.wolfram.com/PrimeConstant.html</a> which cover this particular sequence.</p>