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Old 2018-05-15, 20:35   #12
Till
 
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Could upper bounds (like from https://arxiv.org/pdf/1504.05485.pdf) be helpful?
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Old 2018-05-15, 23:08   #13
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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
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Old 2018-05-16, 11:57   #14
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Quote:
Originally Posted by CRGreathouse View Post
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.
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Old 2018-05-16, 12:10   #15
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Quote:
Originally Posted by ATH View Post
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.
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Old 2018-05-16, 14:17   #16
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Quote:
Originally Posted by Bobby Jacobs View Post
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?
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Old 2018-05-16, 15:28   #17
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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.
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Old 2018-05-16, 15:32   #18
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Quote:
Originally Posted by CRGreathouse View Post
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.
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Old 2018-05-16, 18:08   #19
ATH
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Quote:
Originally Posted by Bobby Jacobs View Post
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 View Post
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.
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Old 2018-05-16, 20:54   #20
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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.)
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Old 2018-05-17, 00:16   #21
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Quote:
Originally Posted by ATH View Post
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.
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Old 2018-05-20, 19:47   #22
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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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