The new record prime gaps

[Bobby Jacobs]

Now, we know two new record prime gaps, 1530 and 1550. There are some interesting things about the new maximal prime gaps.

The merits of all of the last 5 maximal prime gaps are less than the merit of the gap of size 1476. In fact, they are all less than the merit of the 1442 gap. Is that unusual?

The CSG ratios of the last few maximal prime gaps are below 0.8. There were a lot of gaps with CSG ratio above 0.8 before that. The new CSG ratios seem very low. Is that true?

I believe that the next maximal gap after 1550 will have high merit and CSG ratio. It will probably be very big.

[rudy235]

Originally Posted by Bobby Jacobs

Now, we know two new record prime gaps, 1530 and 1550. There are some interesting things about the new maximal prime gaps.

The merits of all of the last 5 maximal prime gaps are less than the merit of the gap of size 1476. In fact, they are all less than the merit of the 1442 gap. Is that unusual?

The CSG ratios of the last few maximal prime gaps are below 0.8. There were a lot of gaps with CSG ratio above 0.8 before that. The new CSG ratios seem very low. Is that true?

I believe that the next maximal gap after 1550 will have high merit and CSG ratio. It will probably be very big.

Hi. No, it is not unusual at all. Of the 80 maximal gaps a little bit over 50% (43) happen to have less merit than the highest merit up to that gap. Al such only 37 of the 80 gaps happen to have a higher merit than all previous.
if a new gap were to be found close to 1.85e19 it would have to be at least 1576 to get a higher merit than 35.31 . In average the Maximal Gaps grow about 14~16 each time, {1476, 1488, 1510, 1526,1530, 1550} and very rarely over 28 in one turn, so it might take a few lucky breaks to be a higher merit than 35.31

[Bobby Jacobs]

I hope the next maximal prime gap is over 1600.

[Bobby Jacobs]

The thing to look at is the value of (g-log²(p)+2*log(p)*log(log(p)))/log(p). An average maximal prime gap will have a value of about 0. A big gap will have a value greater than 0. A small gap will have a value less than 0. The last few maximal prime gaps have low values.

[Bobby Jacobs]

Call the number (g-log²(p)+2*log(p)*log(log(p)))/log(p) the Jacobs value of the prime gap. Then, the gap with the biggest Jacobs value known is the gap of 1132 between 1693182318746371 and 1693182318747503. It has a Jacobs value of 4.3316.

[rudy235]

Originally Posted by Bobby Jacobs

Call the number (g-log²(p)+2*log(p)*log(log(p)))/log(p) the Jacobs value of the prime gap. Then, the gap with the biggest Jacobs value known is the gap of 1132 between 1693182318746371 and 1693182318747503. It has a Jacobs value of 4.3316.

I believe the C-S-G already has the gap of 1132 as the highest in that order.
The Cramér–Shanks–Granville ratio is the ratio of gn / (ln(pn))2 in the case of 1132 it is 0.9206386 for the prime 1693182318746371.

The next highest C-S-G ratio belongs to the gap of 906 discovered by Dr. Nicely in 1996 after prime 218209405436543. The C-S-G ratio is 0.8311

The next in decreasing order is the gap of 766 discovered by Young & Potler in 1989 after prime 19581334192423 The C-S-G ratio is 0.8178

[Bobby Jacobs]

FYI, the gaps of 1530 and 1550 both have Jacobs values of about -2. That is very low. I knew that they seemed like small gaps!