View Single Post
Old 2021-09-11, 17:51   #52
Bobby Jacobs
 
Bobby Jacobs's Avatar
 
May 2018

233 Posts
Default

Quote:
Originally Posted by Bobby Jacobs View Post
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.
I decided to take log2(p) of all of the primes in the maximal prime gaps.

2, 3, 1, 1.0, 1.5849625007211563
3, 5, 2, 1.5849625007211563, 2.321928094887362
7, 11, 4, 2.807354922057604, 3.4594316186372978
23, 29, 6, 4.523561956057013, 4.857980995127573
89, 97, 8, 6.475733430966398, 6.599912842187128
113, 127, 14, 6.820178962415189, 6.988684686772166
523, 541, 18, 9.030667136246942, 9.079484783826816
887, 907, 20, 9.792790294301064, 9.824958740528523
1129, 1151, 22, 10.140829770773001, 10.16867211813223
1327, 1361, 34, 10.373952655370193, 10.410451351503994
9551, 9587, 36, 13.22143607744528, 13.226863716982413
15683, 15727, 44, 13.93691393843731, 13.940955875611515
19609, 19661, 52, 14.259228343849179, 14.263049081612024
31397, 31469, 72, 14.938339094975541, 14.941643713954356
155921, 156007, 86, 17.250455722905738, 17.2512512383879
360653, 360749, 96, 18.46025189899012, 18.46063586999309
370261, 370373, 112, 18.498183071287098, 18.498619405146496
492113, 492227, 114, 18.908630102645986, 18.90896427018037
1349533, 1349651, 118, 20.364028824642652, 20.364154964998352
1357201, 1357333, 132, 20.372202967479005, 20.37234327572016
2010733, 2010881, 148, 20.939290091967138, 20.939396277626003
4652353, 4652507, 154, 22.149529135616888, 22.149576890238947
17051707, 17051887, 180, 24.023412834966237, 24.02342806415923
20831323, 20831533, 210, 24.312251132247198, 24.31226567594332
47326693, 47326913, 220, 25.496150780051064, 25.49615748646031
122164747, 122164969, 222, 26.864252786761337, 26.86425540845062
189695659, 189695893, 234, 27.499111423558197, 27.49911320320056
191912783, 191913031, 248, 27.515875569415407, 27.51587743374218
387096133, 387096383, 250, 28.528116654614454, 28.528117586356245
436273009, 436273291, 282, 28.700655980036846, 28.700656912571894
1294268491, 1294268779, 288, 30.269489783877926, 30.269490104905696
1453168141, 1453168433, 292, 30.436554495809393, 30.4365547857049
2300942549, 2300942869, 320, 31.099577816119524, 31.099578016760077
3842610773, 3842611109, 336, 31.839439703844857, 31.8394398299949
4302407359, 4302407713, 354, 32.002496981951865, 32.002497100656115
10726904659, 10726905041, 382, 33.32051478290403, 33.3205148342804
20678048297, 20678048681, 384, 34.26738097180326, 34.2673809985947
22367084959, 22367085353, 394, 34.38065819502348, 34.38065822043679
25056082087, 25056082543, 456, 34.54444179308049, 34.54444181933634
42652618343, 42652618807, 464, 35.311915256759846, 35.311915272454314
127976334671, 127976335139, 468, 36.897086096100765, 36.897086101376594
182226896239, 182226896713, 474, 37.406944956835446, 37.40694496058811
241160624143, 241160624629, 486, 37.81120341206439, 37.811203414971786
297501075799, 297501076289, 490, 38.11410392914653, 38.11410393152272
303371455241, 303371455741, 500, 38.14229438999324, 38.142294392371014
304599508537, 304599509051, 514, 38.14812265783897, 38.14812266027346
416608695821, 416608696337, 516, 38.599901996646274, 38.599901998433154
461690510011, 461690510543, 532, 38.748135122042186, 38.748135123704586
614487453523, 614487454057, 534, 39.16059259801241, 39.16059259926613
738832927927, 738832928467, 540, 39.42645720880682, 39.42645720986126
1346294310749, 1346294311331, 582, 40.29213096779726, 40.29213096842094
1408695493609, 1408695494197, 588, 40.35749692819665, 40.357496928798845
1968188556461, 1968188557063, 602, 40.84000557906143, 40.8400055795027
2614941710599, 2614941711251, 652, 41.24991592650985, 41.24991592686957
7177162611713, 7177162612387, 674, 42.70655074661664, 42.70655074675213
13829048559701, 13829048560417, 716, 43.65276713583621, 43.6527671359109
19581334192423, 19581334193189, 766, 44.15454430119517, 44.15454430125161
42842283925351, 42842283926129, 778, 45.284100625854215, 45.28410062588041
90874329411493, 90874329412297, 804, 46.368938046534566, 46.36893804654733
171231342420521, 171231342421327, 806, 47.282940127218566, 47.28294012722535
218209405436543, 218209405437449, 906, 47.63270661562181, 47.6327066156278
1189459969825483, 1189459969826399, 916, 50.07922814332398, 50.07922814332509
1686994940955803, 1686994940956727, 924, 50.583377070516875, 50.58337707051766
1693182318746371, 1693182318747503, 1132, 50.588658751647024, 50.58865875164798
43841547845541059, 43841547845542243, 1184, 55.283148252391904, 55.283148252391946
55350776431903243, 55350776431904441, 1198, 55.61945307272224, 55.61945307272227
80873624627234849, 80873624627236069, 1220, 56.16651879039921, 56.16651879039923
203986478517455989, 203986478517457213, 1224, 57.50125113772148, 57.50125113772149
218034721194214273, 218034721194215521, 1248, 57.59733551004153, 57.59733551004153
305405826521087869, 305405826521089141, 1272, 58.08350519916523, 58.08350519916523
352521223451364323, 352521223451365651, 1328, 58.290487730302445, 58.290487730302445
401429925999153707, 401429925999155063, 1356, 58.477925784547146, 58.47792578454716
418032645936712127, 418032645936713497, 1370, 58.53639322594612, 58.53639322594612
804212830686677669, 804212830686679111, 1442, 59.48035496665738, 59.48035496665738
1425172824437699411, 1425172824437700887, 1476, 60.30584258713822, 60.30584258713822
5733241593241196731, 5733241593241198219, 1488, 62.314056782060014, 62.314056782060014
6787988999657777797, 6787988999657779307, 1510, 62.55768993488138, 62.55768993488138
15570628755536096243, 15570628755536097769, 1526, 63.75546100573405, 63.75546100573405
17678654157568189057, 17678654157568190587, 1530, 63.93864225213069, 63.93864225213069
18361375334787046697, 18361375334787048247, 1550, 63.99330792884274, 63.99330792884274
18470057946260698231, 18470057946260699783, 1552, 64.00182219536605, 64.00182219536605
18571673432051830099, 18571673432051831671, 1572, 64.00973762046758, 64.00973762046758

It turns out that there are 3 maximal prime gaps with a log2 within 0.01 of 64. Other than the 2 in the gap between 2 and 3, where 2 is exactly a power of 2, the only primes with a log2 within 0.01 of an integer are 4302407359 and 4302407713, with a log2 of 32.002. That is especially interesting because 32 and 64 are both powers of 2. It is an amazing coincidence!
Bobby Jacobs is offline   Reply With Quote