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-   -   New Maximal Gaps (https://www.mersenneforum.org/showthread.php?t=26924)

 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=SethTro;582464]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.[/QUOTE]

The input page is now working [URL="https://primegaps.cloudygo.com/"]https://primegaps.cloudygo.com/[/URL]

 Bobby Jacobs 2021-07-13 02:31

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[SUP]2[/SUP](p)-(2*ln(p)*ln(ln(p))). I defined the Jacobs value of a gap to be (g-ln[SUP]2[/SUP](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 2[SUP]64[/SUP] + 1.249 * 10[SUP]17[/SUP] ?
This is 5.36 times further from 2[SUP]64[/SUP] than your last gap.

What speed are you getting on the 1080 TI ? Either in time per interval, like time per 10[SUP]12[/SUP] 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 [B][COLOR="Red"]35.4308[/COLOR][/B] 18571673432051830099

Previous Maximal Gaps in order of Merit.[LIST=1]1476 P19 = 1425172824437699411 [B]35.3103[/B][*]1552 PRP20 = 18470057946260698231 34.9844[*]1442 P18 = 804212830686677669 [B]34.9757[/B][*]1550 P20 = 18361375334787046697 34.9439[*]1510 P19 = 6787988999657777797 34.8234[*]1530 P20 = 17678654157568189057 34.5225[*]1526 P20 = 15570628755536096243 34.5312[/LIST]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.:smile:

 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 2[SUP]64[/SUP].

 robert44444uk 2021-07-19 11:24

[QUOTE=CraigLo;583218]I got really lucky and found another.

1572 35.4308 18571673432051830099[/QUOTE]

Astonishing!

 rudy235 2021-07-27 21:06

[QUOTE=CraigLo;583218]I got really lucky and found another.

1572 35.4308 18571673432051830099[/QUOTE]

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

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