Thread: New Maximal Gaps View Single Post
 2021-07-13, 02:31 #25 Bobby Jacobs     May 2018 2·3·43 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)-(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.