mersenneforum.org No way to measure record prime gaps
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 2018-05-15, 20:35 #12 Till     "Tilman Neumann" Jan 2016 Germany 11×43 Posts Could upper bounds (like from https://arxiv.org/pdf/1504.05485.pdf) be helpful?
 2018-05-15, 23:08 #13 rudy235     Jun 2015 Vallejo, CA/. 11111100112 Posts Somewhat unrelated to the thread This are the HIGHEST MERITS in gaps where p<264 All gaps are 2000>GAP>1000 Code:  1476* CFC TOeSilva 2009 35.31 19 1425172824437699411 1442* CFC HrzogTOS 2005 34.98 18 804212830686677669 1550 C?C Be.Nyman 2014 34.94 20 18361375334787046697 1510* CFC Jacobsen 2017 34.82 19 6787988999657777797 1526* CFC ColeStev 2018 34.53 20 15570628755536096243 1530 C?C Be.Nyman 2014 34.52 20 17678654157568189057 1488* CFC ANair_MF 2017 34.45 19 5733241593241196731 1502 CFC Ritschel 2017 34.37 19 9586724781371233277 1490 CFC Jacobsen 2017 34.28 19 7562321651334729221 1454 CFC PardiTOS 2011 34.12 19 3219107182492871783 1506 CFC Ritschel 2018 34.11 20 14950212737555571061 1494 CFC ColeStev 2017 34.05 20 11312044110819306577 1468 CFC RobSmith 2017 34.00 19 5662789033400271433 1486 CFC ColeStev 2017 33.85 20 11596084211508428503 1370* CFC DonKnuth 2006 33.77 18 418032645936712127 1466 CFC Ritschel 2017 33.73 19 7547190328070365163 1482 CFC Ritschel 2017 33.69 20 12685915882452557351 1452 CFC RobSmith 2017 33.63 19 5642831456340753611 1456 CFC Pinho_MF 2017 33.62 19 6446080171984989091 1462 CFC Ritschel 2017 33.61 19 7787601252820494607 1448 CFC Ritschel 2017 33.59 19 5295442011781310951 1440 CFC LLnhardy 2014 33.57 19 4253027105513399527 1356* CFC DonKnuth 2006 33.45 18 401429925999153707 1470 CFC Ritschel 2018 33.39 20 13193289465222402713 1464 CFC ColeStev 2017 33.37 20 11243131368574587263 1434 CFC LLnhardy 2017 33.30 19 5013757591652095753 1358 CFC HrzogTOS 2007 33.29 18 523255220614645319 1430 CFC LLnhardy 2017 33.28 19 4606937813294064947 1380 CFC HrzogTOS 2007 33.27 19 1031501833130243273 1380 CFC HrzogTOS 2007 33.27 19 1031501833130243273 1410 CFC HrzogTOS 2011 33.24 19 2635281932481539903 1418 CFC TOeSilva 2012 33.16 19 3725235533504101511 1450 CFC LMorelli 2017 33.16 19 9808299410025809701 1416 CFC TOeSilva 2012 33.11 19 3750992529339978877 1446 CFC Jacobsen 2017 33.08 19 9656919634106230133 1398 CFC TOeSilva 2011 33.02 19 2424708729726767749 1460 C?C Be.Nyman 2015 32.97 20 16952841674089990313 1426 CFC Ritschel 2017 32.92 19 6508776242818491391 1428 CFC Ritschel 2017 32.89 19 7161997391085310681 1328* CFC TOeSilva 2006 32.87 18 352521223451364323 1364 CFC HrzogTOS 2007 32.87 19 1051140888051230423 1422 CFC Jacobsen 2017 32.87 19 6124339150787745169 1400 CFC PardiTOS 2012 32.80 19 3431657795858378003 1408 CFC LLnhardy 2017 32.73 19 4834864424015986771 1360 CFC HrzogTOS 2008 32.70 19 1153277647303540597 1344 CFC HrzogTOS 2006 32.65 18 753917635380895597 1388 CFC TOeSilva 2011 32.63 19 2975205524123301149 1390 CFC PardiTOS 2012 32.55 19 3492657661005161107 1386 CFC PardiTOS 2012 32.47 19 3449340080274651527 1420 CFC Ritschel 2017 32.46 19 9984196949838014041 1438 C?C Be.Nyman 2014 32.45 20 17554325571496337149 1350 CFC TOeSilva 2008 32.44 19 1180351752204137089 1436 C?C Be.Nyman 2015 32.41 20 17494076304651094403 1374 CFC PardiTOS 2011 32.34 19 2812814235281609869 1424 CFC Jacobsen 2018 32.33 20 13512363187520983367 1404 CFC RobSmith 2017 32.30 19 7529566736915883083 1132* CFC Be.Nyman 1999 32.28 16 1693182318746371 1320 CFC HrzogTOS 2006 32.24 18 605046330029026447 1406 CFC Pinho_MF 2017 32.22 19 8987860411525737317 1990 C?C Spielaur 2013 32.20 27 685245298027055418345996361 1394 CFC Ritschel 2017 32.13 19 6910128698372846693 1750 C?C Spielaur 2016 32.10 24 475135024904107611376237 1936 C?C Spielaur 2016 32.08 27 161023337027994323152086013 1412 CFC Ritschel 2018 32.05 20 13533471241504919417 1396 CFC ColeStev 2017 32.04 19 8382021357433147093 1402 CFC Ritschel 2017 32.04 20 10103695526434940251 1342 CFC TOeSilva 2009 32.03 19 1578169106141187727 1414 CFC RobSmith 2018 32.02 20 15170997176864748337 There are 8 C?C pending, the first 5 in bold red, will be resolved (either way) when the search reaches 2^64-2^32 in 3 months (or a bit more). The other 3 in tourquoise are almost certain to be superseeded if and when the search goes beyond 2^64 or by simple happenning (Helmut Spielauer is still actively seaching in the region of E23 to E27) Another thing to mention is that the Gaps 1530 and 1550 have high merits 34.52 and 34.94 respectively which bodes well for them to become maximal gaps. Code:  1476 Maximal Gap? Yes TOeSilva 2009 35.31 19 1425172824437699411 1442 Maximal Gap? Yes HrzogTOS 2005 34.98 18 804212830686677669 1550 Maximal Gap? Maybe Be.Nyman 2014 34.94 20 18361375334787046697 1510 Maximal Gap? Yes Jacobsen 2017 34.82 19 6787988999657777797 1526 Maximal Gap? Yes ColeStev 2018 34.53 20 15570628755536096243 1530 Maximal Gap? Maybe Be.Nyman 2014 34.52 20 17678654157568189057 1488 Maximal Gap? Yes ANair_MF 2017 34.45 19 5733241593241196731
2018-05-16, 11:57   #14
Bobby Jacobs

May 2018

EA16 Posts

Quote:
 Originally Posted by CRGreathouse CSG is approximately scale invariant. A good value at any size will be around 1 (we think), 1327 to 1361 is just a small gap (not a big gap at a low height) in my interpretation. CSG more than 1 + eps for any fixed eps > 0 should be much more rare than merits 'merely' near 1 which should be fairly common (although we haven't seen it happen yet, ignoring 7 and below).
The problem with CSG is that it is around 0.6 for lower record prime gaps, and it is around 0.8 for higher record prime gaps. It would be good to have a measure that has the same distribution for lower record prime gaps as higher record prime gaps. I believe that 1327 to 1361 is a very big prime gap. Therefore, it should have a greater value than most record prime gaps.

2018-05-16, 12:10   #15
Bobby Jacobs

May 2018

2·32·13 Posts

Quote:
 Originally Posted by ATH For the known big prime gaps GSC is only above 0.8 below 2^64 and it gets smaller and smaller with bigger primes even if those gaps have merit above 35, and it is really tiny for the largest gaps. Any new big gap above 2^64 with GSC above 0.8 or even 0.5 would be something new. Code: Gn Pn Merit Gn/(ln(Pn)^2) Ford-Green-Konyagin-Maynard-Tao 5103138 7.69542115*10^216848 10.22031845 0.00002047 2.12060937 6582144 8.46506984*10^216840 13.18288411 0.00002640 2.73531863 4680156 5.10477651*10^99749 20.37666041 0.00008872 4.50159088 66520 3.29280820*10^815 35.42445941 0.01886489 13.50196237 26892 4.69622677*10^320 36.42056789 0.04932537 16.38392439 26054 5.88832005*10^305 37.00529401 0.05255975 16.80505451 18306 7.04109715*10^208 38.06696007 0.07915948 18.72974074 15900 1.93693327*10^174 39.62015365 0.09872683 20.31243105 13692 3.25418593*10^162 36.59018324 0.09778276 19.07131098 10716 1.83937772*10^126 36.85828850 0.12677617 20.45427745 8382 1.74442287*10^96 37.82412584 0.17068295 22.59742318 8350 2.93703234*10^86 41.93878373 0.21064211 25.84973884 1510 6787988999657777797 34.82336886 0.80309074 43.33266457 1454 3219107182492871783 34.11893253 0.80062005 43.03407437 1476 1425172824437699411 35.31030807 0.84472754 45.22507308 1442 804212830686677669 34.97568651 0.84833471 45.29864017 1370 418032645936712127 33.76518602 0.83218087 44.30979390 1132 1693182318746371 32.28254764 0.92063859 48.34468117 Ford-Green-Konyagin-Maynard-Tao: log X * loglog X * loglogloglog X G(X) > --------------------------------- logloglog X
I am only considering record prime gaps. The gaps at the top of the list are almost certainly not records. However, the new record prime gap of size 1526 has CSG ratio 0.78, which is surprisingly low.

2018-05-16, 14:17   #16
CRGreathouse

Aug 2006

3·1,993 Posts

Quote:
 Originally Posted by Bobby Jacobs The problem with CSG is that it is around 0.6 for lower record prime gaps, and it is around 0.8 for higher record prime gaps. It would be good to have a measure that has the same distribution for lower record prime gaps as higher record prime gaps.
Why do you think it does not?

 2018-05-16, 15:28 #17 rudy235     Jun 2015 Vallejo, CA/. 3×337 Posts There is an element of chance. Not all maximal gaps are created equal. For instance if we look at "gaps between gaps" and compare 2 consecutive maximal gaps we have. Code: 924* CFC Be.Nyman 1999 26.35 16 1686994940955803 CSG 0.7516 1132* CFC Be.Nyman 1999 32.28 16 1693182318746371 CSG 0.9206 Code: 1220* CFC TOeSilva 2003 31.34 17 80873624627234849 CSG 0.8050 1224* CFC TOeSilva 2005 30.71 18 203986478517455989 CSG 0.7705 In the first instance the GAP goes up by 23% and CSG goes up by 22% In the second the GAP goes up by 0.3% and the CSG goes down by 10% The CSG will always go up and down from one gap to the next. I believe it is a safe bet to say that CSGs for maximal gaps will oscillate between 0.76 and 0.93 in the forseable future.
2018-05-16, 15:32   #18
Bobby Jacobs

May 2018

2·32·13 Posts

Quote:
 Originally Posted by CRGreathouse Why do you think it does not?
The CSG ratio has the tendency to get closer to 1 as the size of the primes increases.

2018-05-16, 18:08   #19
ATH
Einyen

Dec 2003
Denmark

2×7×227 Posts

Quote:
 Originally Posted by Bobby Jacobs The problem with CSG is that it is around 0.6 for lower record prime gaps, and it is around 0.8 for higher record prime gaps. It would be good to have a measure that has the same distribution for lower record prime gaps as higher record prime gaps. I believe that 1327 to 1361 is a very big prime gap. Therefore, it should have a greater value than most record prime gaps.
The gap with the highest known merit 41.94 has only a GSC of 0.21.

Since you say GSC tends towards 1 as the size of the primes increases, then show me a prime gap above 2^64 with GSC around 0.8 or even above 0.6.

Quote:
 Originally Posted by Bobby Jacobs I am only considering record prime gaps. The gaps at the top of the list are almost certainly not records. However, the new record prime gap of size 1526 has CSG ratio 0.78, which is surprisingly low.
The 3 top are actually records, but they are the 3 largest gaps in absolute value, not merit.

 2018-05-16, 20:54 #20 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 234438 Posts This reminds me of the solution we used in the lab for comparing 2 results (not enough to run full stats on). Our term of art was RPD (Relative Percent Difference). Since the values could change by quite a bit the RPD was a good test for how good our repeatability was. That and % spike recovery were the 2 main standards. RPD = ABS(S1-S2)/((S1+S2)/2) When working close to the method detection limit, the RPD can grow large with a small absolute difference. 0.1 mg difference when the total value is ~1 mg can screw up a good RPD track record. (The part of the lab that I worked in generally had more tests completed per person and often beat the specialty sections at their own tests WRT RPD. As a unit, for the tests we ran, we had the best average RPD.)
2018-05-17, 00:16   #21
Bobby Jacobs

May 2018

2×32×13 Posts

Quote:
 Originally Posted by ATH The gap with the highest known merit 41.94 has only a GSC of 0.21. Since you say GSC tends towards 1 as the size of the primes increases, then show me a prime gap above 2^64 with GSC around 0.8 or even above 0.6. The 3 top are actually records, but they are the 3 largest gaps in absolute value, not merit.
When I say, "record prime gap," I mean, "maximal prime gap." I am sure that the largest known prime gap is not a maximal prime gap. The CSG ratio of maximal prime gaps approaches 1 as the size of the primes increases. There are no known maximal prime gaps above 2^64, but they probably all have a CSG ratio above 0.8.

 2018-05-20, 19:47 #22 Bobby Jacobs     May 2018 2·32·13 Posts I believe a good measure of record prime gaps would be (g-G(p))/ln(p). This should have the same distribution on all maximal prime gaps. G(p) is the expected largest prime gap between primes up to p.

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