I was wondering if the probability for near-WW primes is linear with A. It feels like it should be (which doesn't say much) and from the findings so far it could be true: 8 for |A|<=100 and 98 for |A|<=1000.

If it is, we could very well find a |A|=1 since we're at 10 % done.
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1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,700,000

The probability for near-primes is linear with A.

The ending point for PrimeGrid's prior Wieferich search on PRPNet is 6.07132e17 and for Wall-Sun-Sun search is 2.6438336e17.

If p = 12e17 (the current leading edge),
the expected number of Wieferich near-primes with |A| <= 1000 is (log(log(12e17)) - log(log(6.07132e17))) * 2001 ~ 33
the expected number of Wall-Sun-Sun near-primes with |A| <= 1000 is (log(log(12e17)) - log(log(2.6438336e17))) * 2001 ~ 74

28 Wieferich and 70 Wall-Sun-Sun near-primes were found with |A| <= 1000.

For |A| <= 10, the expected number of Wieferich and Wall-Sun-Sun near-primes is (33 + 74) / 100 ~ 1.

So far the results match nicely with 13 and 130. I'm not so sure it holds when it comes to A = 0.

We don't look at the actual remainder, but at A = remainder/p, right? So it accomodates for the decreasing prime density at increasing size of the numbers. Thus my feeling is, A = 0 = remainder, will be much less probable and it's probability decreases with increasing size of the number - same as the prime density decreases.

Even more so, if 1800 for |A|<=1000 is really sufficient to produce a Wieferich or WSS prime, then we are 8% there? That seems way too easy to me. If 1.5E18 is 8%, we'd find one within 2E19.

Also I read the density of Wieferich primes is log(log(n)) for 1 to n. Given that 2 are already known, we'd need to go up to 10^1000 to have a good chance to find another one. Even if we chalk one of them up to pure chance, another would be due only within 1...10^100. Far beyond our reach.

Or did I get you wrong?
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1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,700,000

(Of course, we have no proof these heuristics are correct. In theory, Wieferich primes could be frequent or more rare.)

edit: I only saw Jeppe's post about the stats page now, so I adjusted the number accordingly.


I made a table comparing the actual number of |A| in each category with the expected number, iff (oh yeah) 2001:201:21:1 is correct. I only re-distributed the number of near-WW to follow that ratio, this is by no means a calculation of the expected total number of near-WW.

Update

Update

Update

Update 26-04-2021

Update 03-05-2021

Update 10-05-2021

What the prediction doesn't take into account is that the near-WWs are apparently getting more rare with increasing p.

Thanks Ravi, is it still correct to estimate the expected number of near-WW and WW from the actual number of near-WW found? I.e. what I do with that table.

Update 17-05-2021

Yes, the table looks good to me.

Update 03-25-2021

Update 06-01-2021

Update 06-07-2021

Update 06-14-2021

Update 06-21-2021

Update 06-28-2021

Update 07-05-2021

Update 07-12-2021

Update 19-07-2021

Update 26-07-2021

Update 03-08-2021

Update 09-08-2021

Update 17-08-2021

Update 24-08-2021

Update 31-08-2021

A lot of finds this week (ok, 1.5 weeks) and also a WSS with A=-1!

Update 09-09-2021

Update 13-09-2021

Update 20-09-2021

Update 27-09-2021

Update 04-10-2021

You can see that from the ever decreasing "At 18e18 we would reach:" value. It decreases because we find fewer and fewer near hits. Or did you mean something else?

Honestly, I'm not really convinced it makes sense as a forecast (likely it doesn't) since it doesn't take into account that these misses already happened and probability doesn't care what did occur. Just because we found no WWS prime with that many nears doesn't mean we're bound to find one with a specific probability.

Anyway, I think it's a nice statistic that shows how the subproject progresses and it also shows that the "nearness" appears to be statistically distributed as JeppeSN suggested.


Update 11-10-2021

Update 18-10-2021

We reached 2^63 of 2^64 tests. At first glance it looks like we're nearly done, but exponentials being counter-intuitive to many people (like me) and it's just half way done... ;)


Update 05-11-2021

Update 09-11-2021

Update 15-11-2021

Update 22-11-2021

A bit late...

Update 29-11-2021

Things were slow. The GFN-challenge seems to have drained GPU power from WWSS. My 1660 Super will be back soon though, so expect a big speed-up next week.

Update 13-12-2021

No new near-WWSSprimes this week. Not unexpected, since during the challenge the completion rate had dropped from 14k/day to 5k/day. It's back up again though, so unless many computers take a break over the christmas holidays, hopefully this week will fare better again.

Update 20-12-2021

Update 03-01-2022

Last update ist two weeks back, so mind that when looking at the differences. And we reached 1e19. Only 8 quintillion numbers left!

Update 10-01-2022

582 days could be a best case therefore but if loads of people jump on it how dramatically could that come down? I'm playing safe & only parking at 1.1B credit - 2B would take me nearly all year but only assuming enough work!

Yes, it's only the ETA based on last week's numbers. I clarified that in the stats.

Update 17-01-2022

Update 24-01-2022

No new near-hit. High GPU prices are hurting innocent DC-project! The humanity!


Update 01-01-2022

Update 07-02-2022

I think he might just have gotten tired of this or forgot-he was last online 10 days ago on mersenneforum.
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My lucky number is 6219*2^3374198+1

The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.
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The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.

I am confused if this tells us how much the project is finished on an on-going basis.
How, please, can I work out how much is left to be done?
The rate this has picked up recently, WW may not be a viable challenge in September.
(Really I am worried about not getting the next badge)

The leading edge is 11,706.982 P (visible on http://www.primegrid.com/server_status_subprojects.php), which means that the project is just over 65% now.

I am confused if this tells us how much the project is finished on an on-going basis.
How, please, can I work out how much is left to be done?
The rate this has picked up recently, WW may not be a viable challenge in September.
(Really I am worried about not getting the next badge)

The leading edge is now 12,612.271P. The limit of the software for project is 18,000P, so that is when the project will end. So we are at 70% now.

For reference, each work unit advances the project by .003P, so my 3070 is advancing the project by about .270P each day.

I like to use the force in these calculations - oh crap - I won't get the next badge.
But I am going to go for it.
It is worthwhile - to try.

The leading edge is currently 13,385.296P. This means the project is 72.56%. In the last month, the project has averaged 14590 tasks per day, which means the project has about 231 more days before it will be completed at the current rate. There is about 40.49 billion credit left to be claimed for the project.
____________

The leading edge is currently 13,609.720P. This means the project is 73.78% complete. In the last month, the project has averaged 15660 tasks per day, which means the project has about 205 more days before it will be completed at the current rate. There is about 38.69 billion credit left to be claimed for the project.
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The leading edge is currently 13,799.158P. This means the project is 74.81% complete. In the last month, the project has averaged 16560 tasks per day, which means the project has about 187 more days before it will be completed at the current rate. There is about 37.18 billion credit left to be claimed for the project.
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So, the "∞" in the B column for WW on the home page, isn't really true?

For Wall-Sun-Sun prime, it is time to hand over the problem to mathematicians.
No prime for p < 2⁶⁴... probably an hidden property that could be found: no Wall-Sun-Sun prime exists.

The leading edge is currently 13,966.825P. This means the project is 75.71% complete. In the last month, the project has averaged 16790 tasks per day, which means the project has about 178 more days before it will be completed at the current rate. There is about 35.84 billion credit left to be claimed for the project.
____________

The leading edge is currently 14,149.069P. This means the project is 76.70% complete. In the last month, the project has averaged 17160 tasks per day, which means the project has about 167 more days before it will be completed at the current rate. There is about 34.38 billion credit left to be claimed for the project.
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means the project has about 167 more days before it will be completed at the current rate.

But there is a WW challenge in 41 days. That will change something. If we do 20 times the normal rate during those 3 days, there could be about two months left after the challenge.

The project could end before the turn of the year.

/JeppeSN

The leading edge is currently 14,329.645P. This means the project is 77.68% complete. In the last month, the project has averaged 17640 tasks per day, which means the project has about 156 more days before it will be completed at the current rate. There is about 32.93 billion credit left to be claimed for the project.

The project has been advancing forward by about 1% per week, which means that two different mechanisms for estimating when the project will finish are pretty close to each other. Looks like it will be 5-6 months, ignoring the potentially substantial impact that the challenge will have. If the challenge increases the amount of work done by 20x, then that means that there is really about three months of time left before the project finishes.
____________

The leading edge is currently 14,516.467P. This means the project is 78.69% complete. In the last month, the project has averaged 17330 tasks per day, which means the project has about 151 more days before it will be completed at the current rate. There is about 31.44 billion credit left to be claimed for the project.
____________

The leading edge is currently 14,706.865P. This means the project is 79.73% complete. In the last month, the project has averaged 17430 tasks per day, which means the project has about 143 more days before it will be completed at the current rate. There is about 29.92 billion credit left to be claimed for the project.
____________

The leading edge is currently 14909.302P.
This means the project is 80.82% complete.
In the last month, the project has averaged 17960 tasks per day, which means the project has about 131 more days before it will be completed at the current monthly rate.
In the last day, the project completed 22050 tasks, which means the project has about 107 more days before it will be completed at the current rate.
There is about 28.3 billion credit left to be claimed for the project.

____________

The leading edge is currently 15115.198P.
This means the project is 81.94% complete.
In the last month, the project has averaged 18220 tasks per day, which means the project has about 122 more days before it will be completed at the current monthly rate.
In the last day, the project completed 19050 tasks, which means the project has about 117 more days before it will be completed at the current rate.
There is about 26.65 billion credit left to be claimed for the project.

The next update will be in the middle of the challenge. It will be interesting to see how far the project is catapulted forward by the challenge. My guestimate is that there will be about 60 days of work left after the challenge is over, meaning that we will end at around 92% of the project completed.
____________

The leading edge is currently 15478.615P.
This means the project is 83.91% complete.
In the last month, the project has averaged 20060 tasks per day, which means the project has about 99 more days before it will be completed at the current monthly rate.
In the last day, the project completed 97850 tasks, which means the project has about 20 more days before it will be completed at the current rate. However this is a rate from the challenge, and so it is not representative of when the project will actually finish.
There is about 23.75 billion credit left to be claimed for the project.

____________

The leading edge is currently 16168.447P.
This means the project is 87.65% complete.
In the last month, the project has averaged 31510 tasks per day, which means the project has about 48 more days before it will be completed at the current monthly rate.
In the last day, the project completed 20990 tasks, which means the project has about 72 more days before it will be completed at the current rate.
There is about 18.23 billion credit left to be claimed for the project.

The challenge caused quite the advancement, but there is still likely to be a little more than two months before the project finishes.
____________

The leading edge is currently 16319.773P.
This means the project is 88.47% complete.
In the last month, the project has averaged 31410 tasks per day, which means the project has about 45 more days before it will be completed at the current monthly rate.
In the last day, the project completed 15030 tasks, which means the project has about 94 more days before it will be completed at the current rate.
There is about 17.02 billion credit left to be claimed for the project.

____________

The leading edge is currently 16460.521P.
This means the project is 89.23% complete.
In the last month, the project has averaged 30140 tasks per day, which means the project has about 44 more days before it will be completed at the current monthly rate. The monthly completion rate is still very much skewed by the last challenge.
In the last day, the project completed 14640 tasks, which means the project has about 90 more days before it will be completed at the current rate.
There is about 15.89 billion credit left to be claimed for the project.

____________

The leading edge is currently 16619.257P.
This means the project is 90.09% complete.
In the last month, the project has averaged 29200 tasks per day, which means the project has about 42 more days before it will be completed at the current monthly rate.
In the last day, the project completed 15500 tasks, which means the project has about 79 more days before it will be completed at the current rate.
There is about 14.62 billion credit left to be claimed for the project.

____________

The leading edge is currently 16619.257P. ...
...about 42 more days before it will be completed at the current monthly rate.

So this is the last chance for badges?