mersenneforum.org Covering sets
 User Name Remember Me? Password
 Register FAQ Search Today's Posts Mark Forums Read

 2016-04-08, 17:11 #12 robert44444uk     Jun 2003 Oxford, UK 77E16 Posts Here's some mods to provide a cover using only the primes to 631. 223 covers the remaining place. [0,2],[0,3],[1,5],[4,7],[8,11],[9,23],[42,47],[6,19],[34,41],[3,17],[9,13],[7,31],[20,29],[32,73],[69,83],[76,179],[20,109],[1,43],[18,37],[2,61],[35,71],[43,97],[86,157],[64,67],[101,149],[32,137],[237,251],[38,113],[105,107],[30,53],[11,103],[83,101],[42,197],[101,191],[59,181],[48,89],[16,173],[31,163],[4,139],[32,131],[125,127],[272,349],[294,347],[174,337],[40,59],[250,311],[162,307],[78,293],[266,277],[22,269],[24,257],[22,239],[30,79],[63,233],[22,227],[91,211],[187,199],[112,193],[29,167],[27,151],[393,557],[106,563],[231,547],[207,571],[111,577],[86,541],[531,593],[82,599],[141,601],[345,523],[26,613],[357,617],[2,619],[440,631],[237,521],[58,503],[464,499],[147,491],[171,487],[363,467],[262,463],[32,443],[154,439],[82,433],[171,421],[105,409],[321,401],[104,397],[42,389],[44,383],[189,373],[206,367],[213,359],[38,353],[222,317],[250,281],[114,271],[132,263],[3,607],[56,587],[41,569],[51,509],[14,479],[29,461],[73,457],[76,449],[40,431],[15,419],[3,379],[31,331],[21,313],[68,283],[8,241],[211,229]
 2016-04-08, 19:52 #13 mart_r     Dec 2008 you know...around... 2·311 Posts These are nice improvements! Good to see you're getting somewhere. I've taken some time today to take another approach by looking at all the primes that cover only one number, then trying to cover more with another, possibly bigger prime. My first try resulted in a complete coverage with 114 primes (which is again an improvement of 1): offset number n = Code: 2059627222371271396519236736211588692285997827008862576929928819518835488999195543537304317306229873146376143298809029637870475929224367647409662713946206758724094400231417481121708869585830338699805179142485093942365998884778749291890604517931817749743019332 p = 619# / (523*599) * (677*859) For every integer x $\in$ (n+0, n+2310), gcd(x, p) > 1 Actually, this is true for every x $\in$ (n-4, n+2314), so another approach could be to not look at a whole 2310-range, but only 2300 or thereabouts and then see if enough of the bordering numbers are also divisible by the small primes. BTW: According to Pintz, the lower bound should be somewhere near p = 569# (i.e. another 10 primes less), so there's still some work to be done
 2016-04-08, 21:56 #14 robert44444uk     Jun 2003 Oxford, UK 2×7×137 Posts Hah, you beat me to 114 primes, but I think the following result is clean - the first 114 with max prime pi(114)=619 - this was sitting in my results - what you see leaves one integer uncovered and there is one prime <619 that can fill it. My program crashes when it gets a result rather than announcing it with bells and whistles. [0,2],[0,3],[3,5],[0,7],[1,11],[14,23],[24,47],[20,31],[7,19],[2,17],[16,41],[26,43],[74,79],[4,13],[14,73],[2,29],[2,109],[0,37],[45,61],[13,163],[17,71],[25,97],[68,157],[46,67],[83,149],[39,137],[51,83],[219,251],[51,113],[0,53],[87,107],[208,227],[16,103],[65,101],[147,199],[24,197],[96,191],[30,89],[41,181],[138,179],[88,173],[70,139],[14,131],[31,127],[27,59],[232,311],[60,293],[67,271],[88,269],[116,263],[12,241],[119,239],[45,233],[109,229],[71,211],[122,193],[15,167],[62,151],[201,541],[2,547],[123,557],[303,563],[75,569],[260,571],[129,577],[334,523],[273,593],[64,599],[278,601],[578,607],[38,613],[339,617],[435,619],[400,521],[147,509],[267,503],[327,499],[273,491],[300,479],[345,467],[244,463],[92,457],[243,449],[93,443],[267,439],[261,433],[282,431],[129,421],[105,419],[152,401],[375,397],[255,389],[110,383],[233,359],[235,349],[286,307],[263,281],[250,257],[69,587],[20,487],[59,461],[4,409],[30,379],[58,373],[70,367],[21,353],[99,347],[47,337],[54,331],[9,317],[51,313],[28,283],[264,277] Last fiddled with by robert44444uk on 2016-04-08 at 22:05
 2016-04-09, 09:54 #15 robert44444uk     Jun 2003 Oxford, UK 2×7×137 Posts My algorithm does not allow me to get back to the 114 prime solution posted above without crashing the machine, but the following solution using the primes to pi(113)=617 leave only 1 position uncovered, hence 114 primes for total cover. [0,2],[0,3],[3,5],[0,7],[1,11],[14,23],[24,47],[20,31],[7,19],[2,17],[16,41],[26,43],[74,79],[4,13],[14,73],[2,29],[2,109],[0,37],[45,61],[13,163],[17,71],[25,97],[68,157],[46,67],[83,149],[39,137],[51,83],[219,251],[51,113],[0,53],[87,107],[208,227],[16,103],[65,101],[147,199],[24,197],[96,191],[30,89],[41,181],[138,179],[88,173],[70,139],[14,131],[31,127],[27,59],[232,311],[60,293],[67,271],[88,269],[116,263],[12,241],[119,239],[45,233],[109,229],[71,211],[122,193],[15,167],[62,151],[201,541],[2,547],[123,557],[303,563],[75,569],[260,571],[129,577],[334,523],[273,593],[64,599],[278,601],[578,607],[38,613],[339,617],[400,521],[147,509],[267,503],[327,499],[273,491],[300,479],[345,467],[244,463],[92,457],[243,449],[93,443],[267,439],[261,433],[282,431],[129,421],[105,419],[4,409],[152,401],[375,397],[255,389],[110,383],[316,373],[233,359],[235,349],[286,307],[263,281],[66,587],[20,487],[59,461],[30,379],[40,367],[0,353],[99,347],[47,337],[54,331],[9,317],[51,313],[24,283],[3,277],[207,257],[210,223] The solution is improved to 113 primes using 727 as a cover for positions 521 and 1975 rather than any of 223, 257, 277, 283, 313, 317, 331, 337, 347, 353, 367, 379, 461, 487, or 587. Last fiddled with by robert44444uk on 2016-04-09 at 10:16
 2017-01-04, 12:39 #16 robert44444uk     Jun 2003 Oxford, UK 2·7·137 Posts Robert Gerbicz has posted a 111 prime solution: http://www.mersenneforum.org/showthr...t=21826&page=4 Last fiddled with by robert44444uk on 2017-01-04 at 12:39

 Similar Threads Thread Thread Starter Forum Replies Last Post carpetpool Abstract Algebra & Algebraic Number Theory 1 2017-12-28 12:48 MattcAnderson Miscellaneous Math 3 2017-10-18 00:24 carpetpool carpetpool 1 2017-02-22 08:37 Stargate38 And now for something completely different 13 2017-01-21 11:52 mfgoode Miscellaneous Math 2 2006-04-04 00:18

All times are UTC. The time now is 19:51.

Sun Jan 24 19:51:44 UTC 2021 up 52 days, 16:03, 0 users, load averages: 1.56, 1.63, 1.95