Thoughts on 22699?

[Jwb52z]

I don't understand why there's not more conversation on this forum, so I decided to start one and hope it lights a fire under the members. I haven't been able to participate on the project in years and I was looking at the current project stats and I see that k=22699 only has 61 tests, so I'm wondering if anyone thinks the project really is close to having anotherprime found. What do you think?

[DOSGuy]

It's up to 65 now, so it looks like work is still being issued for that k. At first glance, I assumed that the reason why there are less tests outstanding for that k is because there are less tests for that k in the current range to check. Then I decided to check the number of outstanding tests as a fraction of the total tests issued for each k and discovered the following:

4847: 1/377th
21181: 1/406th
22699: 1/724th
24737: 1/413th
55459: 1/366th
67607: 1/363rd

This means that, for five of the remaining Ks, roughly 1/385th of all of the tests that have ever been issued are outstanding right now. For k=22699, only 1/724th of all tests are currently outstanding, and that's a sufficiently significant anomoly to warrant exploration. Is some bizarre feature of the laws of mathematics causing there to be a much smaller number of Ns to test in the current n range? Perhaps sieving has been particularly successful at eliminating Ns for k=22699? Or is there some other reason why k=22699 seems to be issuing less tests than usual?

The most obvious answer is that about half of the tests for k=22699 have been given to PrimeGrid. Until the number of outstanding tests starts dropping, I think there's no reason to expect that a prime has been found.

[enderak]

I believe the low number just has to do with the number of candidates previously removed by sieving. Primegrid was given entire ranges of n values, they were not given proportionally more n's to test from 22699 or any other k. It also does not have anything to do with an imminent find for 22699.

You can see on the Primegrid SoB page that 22699 and 67607 both have significantly fewer candidates, just like here. Other projects over on PG, such as Riesel and Prime Sierpinski, have similar distributions. It all has to do with which numbers get removed while sieving. I'll have to leave it to others to say whether there is a known reason for this or if it is just "one of those things" - the mathematics behind it is beyond my pay grade, but I do know that it is normal.

[DOSGuy]

It's normal for some Ks to have fewer candidates, but it's not normal for the ratio of outstanding tests to the total number of tests for one K to be half that of every other K. Based on the number of previously completed tests, there should be 122 outstanding tests for k=22699. I'll do the same survey of outstanding tests to completed tests for PrimeGrid:

4478: 1/26th
21181: 1/23rd
22699: 1/25th
24737: 1/28th
55459: 1/24th
67607: 1/28th

Overall, 1/25th of the work that has ever been completed is in progress right now. There is no significant variation between Ks; every K has as many tests in progress as we would expect based on the number of tests that were completed for that K in the past. It's only on SoB that one K is currently significantly underrepresented compared to its historical distribution, which is much less likely to be the result of random chance given that we've been assigning tests for over 10 years! There has to be some explanation for why less tests are being assigned for k=22699. It could be that sieving has been doubly successful on that K compared to other Ks recently, but the discrepancy is certainly noteworthy given the size of the dataset.

[enderak]

Ah, I was in a hurry and didn't thoroughly read your post, DOSGuy. Thanks for the comparison.

That is interesting. IF there were a prime for 22699, the increase in the queue could be old tests expiring and being placed back in the queue. Just in the past 10 minutes I've seen it jump to 67 then back down to 65 while the other queues stayed steady, which I think would be indicative of expired tests being put back into queue then quickly handed back out. But there's been plenty of other speculations of upcoming primes over the years, and I've learned not to get too excited.

P.S. I have learned there is a reason why some k have more candidates removed by sieving - the Proth weight. Guess that counts as the new thing I learned today.

EDIT: According to the Google cached page from June 20, the ratio was 1/573, so it is getting smaller while the rest remain fairly consistant (mid- to high-300's). Very interesting!

[Zuzu]

Ah, I was in a hurry and didn't thoroughly read your post, DOSGuy. Thanks for the comparison.

That is interesting. IF there were a prime for 22699, the increase in the queue could be old tests expiring and being placed back in the queue. Just in the past 10 minutes I've seen it jump to 67 then back down to 65 while the other queues stayed steady, which I think would be indicative of expired tests being put back into queue then quickly handed back out. But there's been plenty of other speculations of upcoming primes over the years, and I've learned not to get too excited.

P.S. I have learned there is a reason why some k have more candidates removed by sieving - the Proth weight. Guess that counts as the new thing I learned today.

EDIT: According to the Google cached page from June 20, the ratio was 1/573, so it is getting smaller while the rest remain fairly consistant (mid- to high-300's). Very interesting!

The Proth weight of k=22699 is 0.041 vs 0.035 for 67607, that is consistent with the total nb of tests in SoB thusfar (~47000 for 22699 vs ~41000 for 67607), but not with the nb of pending tests. I've remarked that #pending(22699) is significantly less than #pending(67607), for months, it's not recent. I don't know where it comes from ; otoh, in Primegrid, the respective numbers of pending tests are as expected from their Proth weigts as well as total nb of tests. Maybe there have been specific extra sieving for 22699 in SoB but even with that, it would be very difficult to obtain a k=22699 queue less populated than the k=67607 queue. Moreover, the "max in progress" value for 22699 is consistent with the respective values for the other candidates, both in SoB and Primegrid, showing the queue has not been left behind. Therefore:
- I don't believe in any imminent prime discovery for 22699 (even if such a discovery would be great news for the project, eliminating a low-weighted candidate );
- either there is an error in counting the number of pending tests, specifically applying to 22699;
- or, many valid tests, a significant fraction of the total in progress, have been purely and simply lost or forgotten for the SoB k=22699 queue, for unknown reasons.
So I think, an investigation is necessary: are there lost or forgotten tests in SoB for 22699 or not ?

[Jwb52z]

Well, now 67607 has gone below 100, so I wonder if that's another possibility now.

[engracio]

Or more than likely, since 67607 and 22699 are the two lightest weight prime left. They would reach the equilibrium line first as all of the dropped test were picked up for testing when more people left SOB beginning of the year.

I think we came down from 90's to 50's on daily test completed the last six months. I hope nobody lost any wu's but I doubt it. Just my pov.

[rpertusio]

I dunno what any of you guys are talking about, just lurking here.

[Death]

I think we came down from 90's to 50's on daily test completed the last six months.

they become even larger.