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        riddles >> medium >> Wide prime gaps
        
(Message started by: JocK on May 31^st, 2006, 2:29pm)

Title: Wide prime gaps
Post by JocK on May 31^st, 2006, 2:29pm

Take two subsequent primes p and q, and calculate their gap-value, defined as the ratio between the square of their gap and the logarithm of their sum:

gap(p, q) = Sqrt(|p - q|) / ln(p + q)

We have for instance:

gap(3, 5) = 0.680
gap(7, 11) = 0.692

Finding larger gap-values turns out not to be so easy. The smallest subsequent primes with a gap between them such that the gap-value exceeds 0.700 are 1327 and 1361:

gap(1327, 1361) = 0.738

Easy: -- Can you find larger gap-values?

(Very) Hard: -- What do you think: can arbitrary large gap-values be reached? Or could there be a theoretical maximum?

Title: Re: Wide prime gaps
Post by SMQ on May 31^st, 2006, 6:22pm

Easy -- some more gap leaders:
[hide]gap(31397, 31469)[/hide] = 0.767985
[hide]gap(370261, 370373)[/hide] = 0.783041
[hide]gap(2010733, 2010881)[/hide] = 0.799985
[hide]gap(20831323, 20831533)[/hide] = 0.825949

Assuming my block-wise prime sieve is working, I don't bleieve there is another gap leader pair < 10¹⁰. But it's a prime sieve I wrote myself, using an arbitrary precision integer library I wrote myself, so I wouldn't stake my reputation on that result just yet. ;)

[edit]Update: still under the assumption my program is actually working: [hide]gap(25056082087,25056082543)[/hide] = 0.866733 is the only other gap leader < 6.5*10¹⁰[/edit]

--SMQ

Title: Re: Wide prime gaps
Post by towr on Jun 1^st, 2006, 3:57am

after 90874329411493 there's a gap with value ~0.863

Title: Re: Wide prime gaps
Post by JohanC on Jun 1^st, 2006, 10:21am

or 804212830686677669 with gap value ~0.9058... ?

Title: Re: Wide prime gaps
Post by SMQ on Jun 1^st, 2006, 11:34am

After doing some digging, it seems that "gap value" is an even more severe measure than merit (http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm). As such, I believe the largest known "gap value" is:

gap(1693182318746371, 1693182318747503) ~= .9409

My research would also appear to show that my program is indeed working (my sequence of largest gaps matches Sloane's (http://www.research.att.com/~njas/sequences/A005250)), but I may as well stop searching by hand as the table reproduced at Mathworld (http://mathworld.wolfram.com/PrimeGaps.html) reaches several orders of magnitude beyond my piddling attempt ;)

The complete list of known gap value leaders would seem to be:

gap(2, 3) = 0.621
gap(3, 5) = 0.680
gap(7, 11) = 0.692
gap(1327, 1361) = 0.738
gap(31397, 31469) = 0.768
gap(370261, 370373) = 0.783
gap(2010733, 2010881) = 0.800
gap(20831323, 20831533) = 0.826
gap(25056082087, 25056082543) = 0.867
gap(2614941710599, 2614941711251) = .872
gap(19581334192423, 19581334193189) = .884
gap(218209405436543, 218209405437449) = .893
gap(1693182318746371, 1693182318747503) = .941

with no further gap value leaders < approx. 3*10¹⁷ (and very unlikely < approx. 4*10¹⁸).

[edit]And with a little more digging, it would appear that definitively answering the (Very) Hard portion of the riddle would involve proving (or disproving) Cramer's conjecture (http://en.wikipedia.org/wiki/Cramer's_conjecture) (the "gap value" metric is slightly less than the sqrt of Cramer's metric: |p - q| / ln(p)²), which would likely involve proving the Riemann hypothesis, among other things.

Nice try, JocK. ;D[/edit]

--SMQ

Title: Re: Wide prime gaps
Post by JocK on Jun 1^st, 2006, 3:01pm

on 06/01/06 at 11:34:35, SMQ wrote:

[..] it would appear that definitively answering the (Very) Hard portion of the riddle would involve proving (or disproving) Cramer's conjecture (http://en.wikipedia.org/wiki/Cramer's_conjecture) (the "gap value" metric is slightly less than the sqrt of Cramer's metric: |p - q| / ln(p)²), which would likely involve proving the Riemann hypothesis, among other things.

Nice try, JocK. ;D

If I say "very hard", I mean very hard! :P