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wu :: forums    riddles    medium (Moderators: Eigenray, towr, Grimbal, ThudnBlunder, Icarus, SMQ, william wu)    Wide prime gaps « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Wide prime gaps  (Read 637 times) JocK Uberpuzzler     Gender: Posts: 877 Wide prime gaps   « on: May 31st, 2006, 2:29pm » Quote Modify 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?       « Last Edit: May 31st, 2006, 2:50pm by JocK » IP Logged solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it. xy - y = x5 - y4 - y3 = 20; x>0, y>0. SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: Wide prime gaps   « Reply #1 on: May 31st, 2006, 6:22pm » Quote Modify Easy -- some more gap leaders: gap(31397, 31469) = 0.767985 gap(370261, 370373) = 0.783041 gap(2010733, 2010881) = 0.799985 gap(20831323, 20831533) = 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: gap(25056082087,25056082543) = 0.866733 is the only other gap leader < 6.5*1010[/edit]   --SMQ « Last Edit: Jun 1st, 2006, 3:50am by SMQ » IP Logged --SMQ towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Wide prime gaps   « Reply #2 on: Jun 1st, 2006, 3:57am » Quote Modify after 90874329411493 there's a gap with value ~0.863 IP Logged Wikipedia, Google, Mathworld, Integer sequence DB JohanC Senior Riddler     Posts: 460 Re: Wide prime gaps   « Reply #3 on: Jun 1st, 2006, 10:21am » Quote Modify or 804212830686677669 with gap value ~0.9058... ? IP Logged SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: Wide prime gaps   « Reply #4 on: Jun 1st, 2006, 11:34am » Quote Modify After doing some digging, it seems that "gap value" is an even more severe measure than merit.  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),  but I may as well stop searching by hand as the table reproduced at Mathworld 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 (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. [/edit]     --SMQ « Last Edit: Jun 1st, 2006, 12:20pm by SMQ » IP Logged --SMQ JocK Uberpuzzler     Gender: Posts: 877 Re: Wide prime gaps   « Reply #5 on: Jun 1st, 2006, 3:01pm » Quote Modify on Jun 1st, 2006, 11:34am, SMQ wrote: [..] it would appear that definitively answering the (Very) Hard portion of the riddle would involve proving (or disproving) 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.  If I say "very hard", I mean very hard!   IP Logged solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it. xy - y = x5 - y4 - y3 = 20; x>0, y>0. 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