mersenneforum.org  

Go Back   mersenneforum.org > Math Stuff > Computer Science & Computational Number Theory

Reply
 
Thread Tools
Old 2016-04-08, 17:11   #12
robert44444uk
 
robert44444uk's Avatar
 
Jun 2003
Oxford, UK

77E16 Posts
Default

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]
robert44444uk is offline   Reply With Quote
Old 2016-04-08, 19:52   #13
mart_r
 
mart_r's Avatar
 
Dec 2008
you know...around...

2·311 Posts
Default

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
mart_r is offline   Reply With Quote
Old 2016-04-08, 21:56   #14
robert44444uk
 
robert44444uk's Avatar
 
Jun 2003
Oxford, UK

2×7×137 Posts
Default

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
robert44444uk is offline   Reply With Quote
Old 2016-04-09, 09:54   #15
robert44444uk
 
robert44444uk's Avatar
 
Jun 2003
Oxford, UK

2×7×137 Posts
Default

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
robert44444uk is offline   Reply With Quote
Old 2017-01-04, 12:39   #16
robert44444uk
 
robert44444uk's Avatar
 
Jun 2003
Oxford, UK

2·7·137 Posts
Default

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
robert44444uk is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Covering sets for a^n-1 carpetpool Abstract Algebra & Algebraic Number Theory 1 2017-12-28 12:48
sets of 3 primes MattcAnderson Miscellaneous Math 3 2017-10-18 00:24
Polynomial Coefficients Integer Sets carpetpool carpetpool 1 2017-02-22 08:37
Does 2^n-n-2 have a covering set? Stargate38 And now for something completely different 13 2017-01-21 11:52
Julia Sets 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

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.