Problems & Puzzles: Conjectures

Conjecture 66. Gaps between consecutive twin pairs

Luis Rodríguez sent the following conjecture:

...I search for a formula that would give the approximate value of the maximum gap between twin primes. I propose  0.45 (Log N)^3 . This produces acceptable numbers. The following is a table of mine:

GAP         N          0.45 (LOG N)^3
210        5879             294
630       62927             607
1452      851801            1146
1512     2870471            1480
1722     9925709            1882
2256    30754487            2306
2634    78796691            2705

N represents the first prime of the pair of twins where the gap appears.

Let's name the two pairs of consecutive twins this was: {(p1,p1+2; p2,p2+1}. Then, according to Luis, N=p1, Gap=p2-p1, and his conjecture is this one:

Gap ~ k.(ln(p1))3, k ~ 0.45

Q1. Can you justify this Conjecture or suggest a better one?

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Note: I believe that the formula of Luis is related to the so called "champion gaps" (a gap is a champion when it first occurs and no other gap before is larger than it)

Contribution came from Carlos Rivera

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Carlos Rivera wrote

Here are my own computations for all the champion gaps less than 6032 (p1<2^32):

 Gap p1 p1+2 p2 p2+2 Gap/(ln(p1)^3) 8 3 5 11 13 6.033 12 17 19 29 31 0.528 18 41 43 59 61 0.351 30 71 73 101 103 0.387 36 311 313 347 349 0.190 72 347 349 419 421 0.360 150 659 661 809 811 0.549 168 2381 2383 2549 2551 0.357 210 5879 5881 6089 6091 0.321 282 13397 13399 13679 13681 0.329 372 18539 18541 18911 18913 0.392 498 24419 24421 24917 24919 0.483 630 62297 62299 62927 62929 0.468 924 187907 187909 188831 188833 0.516 930 687521 687523 688451 688453 0.383 1008 688451 688453 689459 689461 0.415 1452 850349 850351 851801 851803 0.570 1512 2868959 2868961 2870471 2870473 0.460 1530 4869911 4869913 4871441 4871443 0.419 1722 9923987 9923989 9925709 9925711 0.412 1902 14656517 14656519 14658419 14658421 0.423 2190 17382479 17382481 17384669 17384671 0.473 2256 30752231 30752233 30754487 30754489 0.440 2832 32822369 32822371 32825201 32825203 0.546 2868 96894041 96894043 96896909 96896911 0.461 3012 136283429 136283431 136286441 136286443 0.458 3102 234966929 234966931 234970031 234970033 0.433 3180 248641037 248641039 248644217 248644219 0.440 3480 255949949 255949951 255953429 255953431 0.480 3804 390817727 390817729 390821531 390821533 0.491 4770 698542487 698542489 698547257 698547259 0.565 5292 2466641069 0.523 6030 4289385521 0.553

and here is the corresponding graph.

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