20210701, 20:12  #23 
"Seth"
Apr 2019
19·23 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 20210701 at 20:13 
20210707, 06:41  #24  
Jun 2003
Suva, Fiji
2^{3}×3×5×17 Posts 
Quote:


20210713, 02:31  #25 
May 2018
2·3^{3}·5 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 ln^{2}(p)(2*ln(p)*ln(ln(p))). I defined the Jacobs value of a gap to be (gln^{2}(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, 2.774068078741657E4 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. 
20210715, 04:24  #26 
Mar 2021
59 Posts 
I got really lucky and found another.
1572 35.4308 18571673432051830099 
20210715, 10:33  #27 
"Seth"
Apr 2019
1B5_{16} Posts 
Amazing work Craig!
It's great to see the lower bound pushed upwards! 
20210715, 11:36  #28 
Einyen
Dec 2003
Denmark
2^{3}·3^{2}·47 Posts 
Gratz again!
Did you really test everything up to 2^{64} + 1.249 * 10^{17} ? This is 5.36 times further from 2^{64} than your last gap. What speed are you getting on the 1080 TI ? Either in time per interval, like time per 10^{12} or whatever, or interval per time, how far per hour or per day? 
20210715, 12:07  #29 
Jun 2015
Vallejo, CA/.
3×13×29 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.
Congratulations. Edit: The second gap in the list 1552 gap is not yet proven to be maximal. Last fiddled with by rudy235 on 20210715 at 12:22 
20210716, 04:25  #30 
Mar 2021
59 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.

20210717, 11:14  #31 
May 2018
2×3^{3}×5 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 2^{64}.

20210719, 11:24  #32 
Jun 2003
Suva, Fiji
3770_{8} Posts 

20210727, 21:06  #33 
Jun 2015
Vallejo, CA/.
3·13·29 Posts 

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