Covering sets - Page 2

robert44444uk · 2016-04-08, 17:11

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]

mart_r · 2016-04-08, 19:52

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

robert44444uk · 2016-04-08, 21:56

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]

robert44444uk · 2016-04-09, 09:54

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.

robert44444uk · 2017-01-04, 12:39

Robert Gerbicz has posted a 111 prime solution:

http://www.mersenneforum.org/showthr...t=21826&page=4

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# / (523599) (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

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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]