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Old 2018-09-01, 19:35   #34
Bobby Jacobs
 
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In this article, they talk about rescaling gaps. There is the formula (g-T)/a. I believe that this is what I want. However, the values for maximal prime gaps are not given in the article. This article talks about primes that are r mod q. Regular prime gaps are between primes that are 0 mod 1. Therefore, a(p)=p/li(p) and T(p)=(p/li(p))(2*log(li(p))-log(p)). Then, the value is (g-((p/li(p))(2*log(li(p))-log(p))))/(p/li(p)). What are the values of this function for the known maximal prime gaps?
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Old 2018-09-01, 20:07   #35
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Quote:
Originally Posted by Bobby Jacobs View Post
Regular prime gaps are between primes that are 0 mod 1.
What does 0 mod 1 mean? Or, what whole number is not 0 mod 1?
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Old 2018-09-03, 03:31   #36
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Quote:
Originally Posted by VBCurtis View Post
What does 0 mod 1 mean? Or, what whole number is not 0 mod 1?
It has to be a typo!

The article was interesting though.
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Old 2018-09-03, 15:17   #37
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Quote:
Originally Posted by VBCurtis View Post
What does 0 mod 1 mean? Or, what whole number is not 0 mod 1?
The article is about maximal gaps between primes that are r mod q. Regular maximal prime gaps are between whole numbers. Since all whole numbers are 0 mod 1, then r=0 and q=1. Therefore, regular maximal prime gaps are between primes that are 0 mod 1.
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Old 2018-09-04, 07:07   #38
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Quote:
Originally Posted by Bobby Jacobs View Post
The article is about maximal gaps between primes that are r mod q. Regular maximal prime gaps are between whole numbers. Since all whole numbers are 0 mod 1, then r=0 and q=1. Therefore, regular maximal prime gaps are between primes that are 0 mod 1.
But ALL primes (except 2) are 1 Mod 2, so that is a stronger restriction. There is only 1 maximal gap that involves prime 2.
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Old 2018-09-11, 21:37   #39
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The greatest gap between primes up to n is about log2(n)-2*log(n)*log(log(n)). Therefore, a good measure would be (g-(log2(p)-2*log(p)*log(log(p))))/log(p)=(g-log2(p)+2*log(p)*log(log(p)))/log(p). Which maximal prime gap has the biggest value of (g-log2(p)+2*log(p)*log(log(p)))/log(p)?

Last fiddled with by Bobby Jacobs on 2018-09-11 at 21:40
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Old 2018-09-12, 12:37   #40
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Quote:
Originally Posted by Bobby Jacobs View Post
The greatest gap between primes up to n is about log2(n)-2*log(n)*log(log(n)).
What makes you say that, and in what sense do you think that that is true?

It's not at all clear to me that we can meaningfully discuss nonleading terms when even the coefficient of the leading term is in doubt. (Some sources aren't even sure of the exponent.)
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Old 2018-10-11, 15:41   #41
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Here are the values of (g-log2(p)+2*log(p)*log(log(p)))/log(p) for the first 30 maximal prime gaps. This measure seems to have the same distribution on all maximal prime gaps.

p1:
2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009

g:
1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148, 154, 180, 210, 220, 222, 234, 248, 250, 282

p2:
3, 5, 11, 29, 97, 127, 541, 907, 1151, 1361, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291

(g-log2(p1)+2*log(p1)*log(log(p1)))/log(p1):
0.01652201917, 0.9099618198, 1.441142842, 1.06365333, 0.2967400124, 1.340824126, 0.2842368037, -0.01113471657, 0.0008777107308, 1.483239756, -0.8055044796, -0.5695678433, -0.04079693315, 1.273888756, 0.1979231021, -0.1949205926, 1.015367394, 0.7377458039, -0.4610135033, 0.5222047529, 1.033263049, 0.1404118759, -0.2170613361, 1.258418936, 0.5201039801, -0.8502142355, -0.8892300766, -0.1730714975, -1.162682203, 0.2623211567

(g-log2(p2)+2*log(p2)*log(log(p2)))/log(p2):
-0.0002774068079, 0.5850019473, 1.019417059, 0.8427693921, 0.2151204219, 1.201433598, 0.2457202101, -0.03652024352, -0.02150721463, 1.44838762, -0.8100577184, -0.5731101098, -0.04431878473, 1.270502656, 0.1971322563, -0.1953011903, 1.01490609, 0.737395807, -0.4611403309, 0.5220568929, 1.033147879, 0.1403614607, -0.2170774769, 1.258402597, 0.5200965832, -0.850217021, -0.8892319752, -0.1730735353, -1.162683196, 0.2623201147

The gap with the highest value is 1327 to 1361. The gap with the lowest value is 387096133 to 387096383.

Last fiddled with by Bobby Jacobs on 2018-10-11 at 15:47
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Old 2019-02-27, 20:24   #42
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Did somebody change the title of this thread? It used to be called "A way to measure record prime gaps". Now, it is "No way to measure record prime gaps". Who vandalized the title? By the way, I have decided to call my prime gap measure (g-log2(p)+2*log(p)*log(log(p)))/log(p) the Jacobs value.
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Old 2019-02-27, 21:54   #43
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Quote:
Originally Posted by Bobby Jacobs View Post
[...]the Jacobs value.
+20 points!

If only the modern-day Inquisition would stop changing the thread titles.
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Old 2021-09-01, 23:25   #44
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Here are the Jacobs values of the maximal prime gaps, assuming that 1552 and 1572 are maximal prime gaps. Here are the values of (g-ln2(p1)+(2*ln(p1)*ln(ln(p1))))/ln(p1).

2, 3, 1, 0.016522019165689444
3, 5, 2, 0.9099618198189632
7, 11, 4, 1.4411428415802423
23, 29, 6, 1.0636533300832
89, 97, 8, 0.29674001242202547
113, 127, 14, 1.3408241256836613
523, 541, 18, 0.2842368036947438
887, 907, 20, -0.011134716571830336
1129, 1151, 22, 8.777107307645034E-4
1327, 1361, 34, 1.4832397557644585
9551, 9587, 36, -0.805504479605833
15683, 15727, 44, -0.5695678432616934
19609, 19661, 52, -0.040796933154394706
31397, 31469, 72, 1.273888756168562
155921, 156007, 86, 0.19792310211263436
360653, 360749, 96, -0.1949205926177167
370261, 370373, 112, 1.0153673944798356
492113, 492227, 114, 0.7377458039370451
1349533, 1349651, 118, -0.46101350333950425
1357201, 1357333, 132, 0.522204752949181
2010733, 2010881, 148, 1.0332630489365533
4652353, 4652507, 154, 0.14041187587213844
17051707, 17051887, 180, -0.21706133606860137
20831323, 20831533, 210, 1.2584189364590443
47326693, 47326913, 220, 0.5201039801153848
122164747, 122164969, 222, -0.8502142355203356
189695659, 189695893, 234, -0.8892300766355324
191912783, 191913031, 248, -0.17307149751020964
387096133, 387096383, 250, -1.1626822025691668
436273009, 436273291, 282, 0.2623211567153048
1294268491, 1294268779, 288, -1.1673900646978592
1453168141, 1453168433, 292, -1.1579264827865645
2300942549, 2300942869, 320, -0.570569934608561
3842610773, 3842611109, 336, -0.6563442333459786
4302407359, 4302407713, 354, -0.025269902422142616
10726904659, 10726905041, 382, -0.27705458702409175
20678048297, 20678048681, 384, -1.2501473160691379
22367084959, 22367085353, 394, -0.9557069916373576
25056082087, 25056082543, 456, 1.4512178992596374
42652618343, 42652618807, 464, 0.8761319628583183
127976334671, 127976335139, 468, -0.7928312072681498
182226896239, 182226896713, 474, -1.1368024066973388
241160624143, 241160624629, 486, -1.133103775767574
297501075799, 297501076289, 490, -1.3230609986763935
303371455241, 303371455741, 500, -0.9765903766825987
304599508537, 304599509051, 514, -0.4537585898504115
416608695821, 416608696337, 516, -0.8961227352095684
461690510011, 461690510543, 532, -0.46926117459445876
614487453523, 614487454057, 534, -0.8689217985944656
738832927927, 738832928467, 540, -0.952780288905108
1346294310749, 1346294311331, 582, -0.4300706735573055
1408695493609, 1408695494197, 588, -0.29140204350454185
1968188556461, 1968188557063, 602, -0.3558643149972103
2614941710599, 2614941711251, 652, 0.9173811142454119
7177162611713, 7177162612387, 674, -0.05745709513145544
13829048559701, 13829048560417, 716, 0.2250408079258608
19581334192423, 19581334193189, 766, 1.2648691256319562
42842283925351, 42842283926129, 778, 0.2904504058456662
90874329411493, 90874329412297, 804, -0.1850961087042459
171231342420521, 171231342421327, 806, -1.2021259403175406
218209405436543, 218209405437449, 906, 1.41838188935369
1189459969825483, 1189459969826399, 916, -1.2297298698345847
1686994940955803, 1686994940956727, 924, -1.5939815003880229
1693182318746371, 1693182318747503, 1132, 4.331590450309763
43841547845541059, 43841547845542243, 1184, -0.1292269969947945
55350776431903243, 55350776431904441, 1198, -0.17389157190413126
80873624627234849, 80873624627236069, 1220, -0.27108837564179794
203986478517455989, 203986478517457213, 1224, -1.7763241380075547
218034721194214273, 218034721194215521, 1248, -1.2896653554305229
305405826521087869, 305405826521089141, 1272, -1.2753733168794827
352521223451364323, 352521223451365651, 1328, -0.13791061617203643
401429925999153707, 401429925999155063, 1356, 0.32401793693096304
418032645936712127, 418032645936713497, 1370, 0.5971215852470018
804212830686677669, 804212830686679111, 1442, 1.1853125062362995
1425172824437699411, 1425172824437700887, 1476, 0.9753154533071884
5733241593241196731, 5733241593241198219, 1488, -1.2112885884477058
6787988999657777797, 6787988999657779307, 1510, -0.9991650401801192
15570628755536096243, 15570628755536097769, 1526, -2.0836332971905804
17678654157568189057, 17678654157568190587, 1530, -2.213542051864597
18361375334787046697, 18361375334787048247, 1550, -1.8283253937088773
18470057946260698231, 18470057946260699783, 1552, -1.793526663263014
18571673432051830099, 18571673432051831671, 1572, -1.3523184236097123

Here are the values of (g-ln2(p2)+(2*ln(p2)*ln(ln(p2))))/ln(p2).

2, 3, 1, -2.774068078741657E-4
3, 5, 2, 0.5850019473393445
7, 11, 4, 1.0194170587459925
23, 29, 6, 0.8427693920801025
89, 97, 8, 0.2151204218932354
113, 127, 14, 1.2014335975957555
523, 541, 18, 0.24572021013679826
887, 907, 20, -0.03652024352158025
1129, 1151, 22, -0.02150721463226403
1327, 1361, 34, 1.448387619815254
9551, 9587, 36, -0.8100577183892048
15683, 15727, 44, -0.5731101097730286
19609, 19661, 52, -0.044318784730949315
31397, 31469, 72, 1.270502655772665
155921, 156007, 86, 0.19713225629640171
360653, 360749, 96, -0.19530119032678306
370261, 370373, 112, 1.0149060902113427
492113, 492227, 114, 0.7373958069907232
1349533, 1349651, 118, -0.4611403308640233
1357201, 1357333, 132, 0.5220568928672134
2010733, 2010881, 148, 1.033147878685969
4652353, 4652507, 154, 0.14036146068116476
17051707, 17051887, 180, -0.21707747686074932
20831323, 20831533, 210, 1.258402597454145
47326693, 47326913, 220, 0.5200965832122922
122164747, 122164969, 222, -0.8502170210370023
189695659, 189695893, 234, -0.8892319752420521
191912783, 191913031, 248, -0.1730735352653388
387096133, 387096383, 250, -1.1626831960005015
436273009, 436273291, 282, 0.2623201147347242
1294268491, 1294268779, 288, -1.1673904115852733
1453168141, 1453168433, 292, -1.1579267965057205
2300942549, 2300942869, 320, -0.5705701565499374
3842610773, 3842611109, 336, -0.6563443731836452
4302407359, 4302407713, 354, -0.025270036477032205
10726904659, 10726905041, 382, -0.27705464505393207
20678048297, 20678048681, 384, -1.250147345715678
22367084959, 22367085353, 394, -0.9557070199950923
25056082087, 25056082543, 456, 1.4512178681058923
42652618343, 42652618807, 464, 0.8761319444431268
127976334671, 127976335139, 468, -0.7928312132556395
182226896239, 182226896713, 474, -1.1368024109318071
241160624143, 241160624629, 486, -1.133103779054893
297501075799, 297501076289, 490, -1.3230610013550814
303371455241, 303371455741, 500, -0.9765903793850329
304599508537, 304599509051, 514, -0.45375859265074997
416608695821, 416608696337, 516, -0.8961227372483432
461690510011, 461690510543, 532, -0.4692611765107444
614487453523, 614487454057, 534, -0.8689218000292784
738832927927, 738832928467, 540, -0.9527802901109655
1346294310749, 1346294311331, 582, -0.43007067428120926
1408695493609, 1408695494197, 588, -0.2914020442057517
1968188556461, 1968188557063, 602, -0.3558643155112367
2614941710599, 2614941711251, 652, 0.9173811138146628
7177162611713, 7177162612387, 674, -0.0574570952912501
13829048559701, 13829048560417, 716, 0.22504080783702007
19581334192423, 19581334193189, 766, 1.264869125563402
42842283925351, 42842283926129, 778, 0.2904504058143198
90874329411493, 90874329412297, 804, -0.18509610871942694
171231342420521, 171231342421327, 806, -1.202125940325487
218209405436543, 218209405437449, 906, 1.4183818893463445
1189459969825483, 1189459969826399, 916, -1.2297298698358887
1686994940955803, 1686994940956727, 924, -1.5939815003889521
1693182318746371, 1693182318747503, 1132, 4.331590450308519
43841547845541059, 43841547845542243, 1184, -0.1292269969948478
55350776431903243, 55350776431904441, 1198, -0.17389157190416654
80873624627234849, 80873624627236069, 1220, -0.27108837564182264
203986478517455989, 203986478517457213, 1224, -1.7763241380075703
218034721194214273, 218034721194215521, 1248, -1.2896653554305229
305405826521087869, 305405826521089141, 1272, -1.2753733168794827
352521223451364323, 352521223451365651, 1328, -0.13791061617203643
401429925999153707, 401429925999155063, 1356, 0.32401793693094755
418032645936712127, 418032645936713497, 1370, 0.5971215852470018
804212830686677669, 804212830686679111, 1442, 1.1853125062362995
1425172824437699411, 1425172824437700887, 1476, 0.9753154533071884
5733241593241196731, 5733241593241198219, 1488, -1.2112885884477058
6787988999657777797, 6787988999657779307, 1510, -0.9991650401801192
15570628755536096243, 15570628755536097769, 1526, -2.0836332971905804
17678654157568189057, 17678654157568190587, 1530, -2.213542051864597
18361375334787046697, 18361375334787048247, 1550, -1.8283253937088773
18470057946260698231, 18470057946260699783, 1552, -1.793526663263014
18571673432051830099, 18571673432051831671, 1572, -1.3523184236097123

I am suspicious about the low Jacobs values of the last few maximal prime gaps. Are you sure you did not miss any 1600+ gaps?
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