New Maximal Gaps - Page 3

SethTro · 2021-07-01, 20:12

I have internet setup now and I'm busy running ethernet cables all over. Thanks for everyone's patience and DMs checking if I was okay. The server should hopefully be up by Sunday morning.

robert44444uk · 2021-07-07, 06:41

Quote:

Originally Posted by SethTro

I have internet setup now and I'm busy running ethernet cables all over. Thanks for everyone's patience and DMs checking if I was okay. The server should hopefully be up by Sunday morning.

The input page is now working https://primegaps.cloudygo.com/

Bobby Jacobs · 2021-07-13, 02:31

The last few maximal prime gaps seemed smaller than the expected maximal prime gap. Now, I know why. In another post, I conjectured that the maximum prime gap between primes up to p is approximately ln²(p)-(2*ln(p)*ln(ln(p))). I defined the Jacobs value of a gap to be (g-ln²(p)+(2*ln(p)*ln(ln(p))))/ln(p). Then, a Jacobs value of 0 would be an average maximal gap, a value of 1 would be a big maximal gap, and a value of -1 would be a small maximal gap. Here are the Jacobs values of the known maximal gaps.

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

Notice that most of the maximal gaps have a Jacobs value between -2 and 2. A weird anomaly is the gap of 1132, which has a Jacobs value of 4.33. The gaps of 1526 and 1530 are the only maximal gaps with a Jacobs value below -2. The gap of 1550 is almost at -2. If this gap of 1552 is a maximal gap, then it will also have a low Jacobs value. That is why I believe we should have a lot bigger maximal prime gaps.

CraigLo · 2021-07-15, 04:24

I got really lucky and found another.

1572 35.4308 18571673432051830099

SethTro · 2021-07-15, 10:33

Amazing work Craig!

It's great to see the lower bound pushed upwards!

ATH · 2021-07-15, 11:36

Gratz again!

Did you really test everything up to 2⁶⁴ + 1.249 * 10¹⁷ ?
This is 5.36 times further from 2⁶⁴ than your last gap.

What speed are you getting on the 1080 TI ? Either in time per interval, like time per 10¹² or whatever, or interval per time, how far per hour or per day?

rudy235 · 2021-07-15, 12:07

This is so exciting. We now have a "probable maximum gap" with the highest merit of all gaps that have a good chance of being maximal. There are gaps with much higher merits but almost no chance of becoming maximal.

1572 35.4308 18571673432051830099

Previous Maximal Gaps in order of Merit.

1476 P19 = 1425172824437699411 35.3103
1552 PRP20 = 18470057946260698231 34.9844
1442 P18 = 804212830686677669 34.9757
1550 P20 = 18361375334787046697 34.9439
1510 P19 = 6787988999657777797 34.8234
1530 P20 = 17678654157568189057 34.5225
1526 P20 = 15570628755536096243 34.5312

So, should it become a maximal gap it will have the highest merit of all the maximals.

Congratulations.

Edit: The second gap in the list -1552- gap is not yet proven to be maximal.

CraigLo · 2021-07-16, 04:25

Looking for gaps >= 1300 I search about 150E9/sec = 1.3E16/day. That's sieving with about 10k primes and doing 1 Fermat test.

Bobby Jacobs · 2021-07-17, 11:14

Congratulations on finding another gap! It is amazing that there are so many maximal prime gaps so logarithmically close to the binary round number 2⁶⁴.

robert44444uk · 2021-07-19, 11:24

Quote:

Originally Posted by CraigLo

I got really lucky and found another.

1572 35.4308 18571673432051830099

Astonishing!

rudy235 · 2021-07-27, 21:06

Quote:

Originally Posted by CraigLo

I got really lucky and found another.

1572 35.4308 18571673432051830099

i looked for it at the prime gap tables and it was not there. Did you report it?

2021-07-01, 20:12	#23
SethTro "Seth" Apr 2019 433 Posts	I have internet setup now and I'm busy running ethernet cables all over. Thanks for everyone's patience and DMs checking if I was okay. The server should hopefully be up by Sunday morning. Last fiddled with by SethTro on 2021-07-01 at 20:13

2021-07-13, 02:31	#25
Bobby Jacobs May 2018 7·37 Posts	Why I am underwhelmed by the last few maximal prime gaps The last few maximal prime gaps seemed smaller than the expected maximal prime gap. Now, I know why. In another post, I conjectured that the maximum prime gap between primes up to p is approximately ln²(p)-(2ln(p)ln(ln(p))). I defined the Jacobs value of a gap to be (g-ln²(p)+(2ln(p)ln(ln(p))))/ln(p). Then, a Jacobs value of 0 would be an average maximal gap, a value of 1 would be a big maximal gap, and a value of -1 would be a small maximal gap. Here are the Jacobs values of the known maximal gaps. 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 Notice that most of the maximal gaps have a Jacobs value between -2 and 2. A weird anomaly is the gap of 1132, which has a Jacobs value of 4.33. The gaps of 1526 and 1530 are the only maximal gaps with a Jacobs value below -2. The gap of 1550 is almost at -2. If this gap of 1552 is a maximal gap, then it will also have a low Jacobs value. That is why I believe we should have a lot bigger maximal prime gaps.

2021-07-15, 12:07	#29
rudy235 Jun 2015 Vallejo, CA/. 10001001001₂ Posts	This is so exciting. We now have a "probable maximum gap" with the highest merit of all gaps that have a good chance of being maximal. There are gaps with much higher merits but almost no chance of becoming maximal. 1572 35.4308 18571673432051830099 Previous Maximal Gaps in order of Merit. 1476 P19 = 1425172824437699411 35.3103 1552 PRP20 = 18470057946260698231 34.9844 1442 P18 = 804212830686677669 34.9757 1550 P20 = 18361375334787046697 34.9439 1510 P19 = 6787988999657777797 34.8234 1530 P20 = 17678654157568189057 34.5225 1526 P20 = 15570628755536096243 34.5312 So, should it become a maximal gap it will have the highest merit of all the maximals. Congratulations. Edit: The second gap in the list -1552- gap is not yet proven to be maximal. Last fiddled with by rudy235 on 2021-07-15 at 12:22

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2021-07-15, 04:24	#26
CraigLo Mar 2021 59₁₀ Posts	I got really lucky and found another. 1572 35.4308 18571673432051830099

2021-07-15, 10:33	#27
SethTro "Seth" Apr 2019 661₈ Posts	Amazing work Craig! It's great to see the lower bound pushed upwards!

2021-07-15, 11:36	#28
ATH Einyen Dec 2003 Denmark CF1₁₆ Posts	Gratz again! Did you really test everything up to 2⁶⁴ + 1.249 * 10¹⁷ ? This is 5.36 times further from 2⁶⁴ than your last gap. What speed are you getting on the 1080 TI ? Either in time per interval, like time per 10¹² or whatever, or interval per time, how far per hour or per day?

2021-07-16, 04:25	#30
CraigLo Mar 2021 3B₁₆ Posts	Looking for gaps >= 1300 I search about 150E9/sec = 1.3E16/day. That's sieving with about 10k primes and doing 1 Fermat test.

2021-07-17, 11:14	#31
Bobby Jacobs May 2018 7·37 Posts	Congratulations on finding another gap! It is amazing that there are so many maximal prime gaps so logarithmically close to the binary round number 2⁶⁴.