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JarekVolunteer developer Send message
Joined: 28 Dec 08 Posts: 57 ID: 33488 Credit: 10 RAC: 0

When conducting a search like that, when we are waiting very long for a single result, it is good to see the progress  the search is much less frustrating that way.
Assuming we keep track of AP20 and longer, one should expect on average one AP26 to appear in 50006000 AP20+, at least when we stay in reasonably low range.
So the first nonfrustrating goal could be to reach 6000 AP20+ found.
The expected average count of longer AP's at the moment the first AP26 is encountered, to my best knowledge, is as follows:
AP21+ 15001700
AP22+ 400470
AP23+ 110120
AP24+ 2729 (note that many AP24 would break "The Largest Known AP24" record)
AP25+ 5.56 (only one AP25 is known at the moment)
Therefore the searchers can honestly expect, as a byproduct of the AP26 search, to have some 1020 of their names noted as the AP24 or AP25 record holders on the JKA's website:
http://hjem.get2net.dk/jka/math/aprecords.htm
Of course we can be very lucky or unlucky, so those counts can easily be off by a significant factor. Anyway, going with the actual solution count over the above numbers without hiting AP26 gives us the right to complain on bad luck.
Also there is some 11% chance for the first AP26 to be AP27 and about 1% chance to be AP28.
Since the search setup assumes the progression difference to be nondivisible by 29, there is no way to find AP29 (except if the first term is 29, but miracles like that do not happen in real life).



RytisVolunteer moderator Project administrator
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Joined: 22 Jun 05 Posts: 2639 ID: 1 Credit: 21,300,693 RAC: 6,865

And here we have the counts of progressions that PG has found to date: http://www.primegrid.com/stats_ap26.php
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I see an AP24 was found.
Are the highest AP finders going to be shown or anything? (basicly I'm wondering of those shown, how many and/or which ones are 'mine')
Not to make your life difficult, just wondering.
:D
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JarekVolunteer developer Send message
Joined: 28 Dec 08 Posts: 57 ID: 33488 Credit: 10 RAC: 0

This AP24 is a rediscovey. It was found by ksysju, the AP26 Search current leader. I deduced this from informations available publicly.
On the way of the search there are 3 more known AP24 and one AP25.
There are 5 more known AP24 which will be skipped by the search.
The 2nd known AP24 should be rediscovered in a day or two, unless its WU will be somehow delayed.
The only known AP25 should be rediscovered around 100,000th WU computed.
2 more AP24 should come with WU count around 160,000170,000 and 260,000270,000 respectively.
Personally I am in favor of publishing all new AP24 or longer, discovered during the search, as they are rare, hence interesting by itself.
Note that as the search will be moving to a higher range, the frequency of AP's found will be decreasing. The density of primes around, say, 10^17 is only by about 6% lower than around 10^16. But at the level of AP24AP26 this translates to a factor of 45. At this moment the count of AP's found may be boosted by small AP's, which appear at the beginning of the search.



JarekVolunteer developer Send message
Joined: 28 Dec 08 Posts: 57 ID: 33488 Credit: 10 RAC: 0

If I am reading all the data correctly, another AP24 has been rediscovered, this time by Vato. 


JarekVolunteer developer Send message
Joined: 28 Dec 08 Posts: 57 ID: 33488 Credit: 10 RAC: 0

The AP Records wesite maintained by Jens Kruse Andersen now records all known AP24 and longer. Therefore any new AP24+ will be recorded there together with the name of its discoverer.
Also please note that the site has been moved to
http://users.cybercity.dk/~dsl522332/math/aprecords.htm 



In AP26 Stats threads:
Expected AP's to be found
AP26  1
AP25  5.56
AP24  2729
AP23  110120
AP22  400470
AP21  15001700
AP20  50006000
Current statistics:
AP25  4 : In bounds
AP24  28 : In bounds
AP23  113 : In bounds
AP22  467 : In bounds
AP21  [b]1995[/b] : Goal widely surpassed
AP20  [b]7177[/b] : Goal widely surpassed
Good jobs...
Should we raise the upper bounds limits?
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Badge Score: 1*2 + 10*5 + 4*6 + 2*7 + 2*8 + 1*9 = 115 



In AP26 Stats threads:
Expected AP's to be found
AP26  1
AP25  5.56
AP24  2729
AP23  110120
AP22  400470
AP21  15001700
AP20  50006000
Current statistics:
AP25  4 : In bounds
AP24  28 : In bounds
AP23  113 : In bounds
AP22  467 : In bounds
AP21  [b]1995[/b] : Goal widely surpassed
AP20  [b]7177[/b] : Goal widely surpassed
Good jobs...
Should we raise the upper bounds limits?
Yes I would very much appreciate raising the upper bound limits :) Thanks guys.
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John M. Johnson "Novex" 



With all predictions but AP25 surpassed, one must ask the question:
Does this mean that the AP26 is due any time now, or does this mean that we were just wildly off in our predictions?
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JarekVolunteer developer Send message
Joined: 28 Dec 08 Posts: 57 ID: 33488 Credit: 10 RAC: 0

With all predictions but AP25 surpassed, one must ask the question:
Does this mean that the AP26 is due any time now, or does this mean that we were just wildly off in our predictions?
Neither. We cannot expect AP26 any time and also the predictions are off only slightly.
The problem is to understand correctly what all the predictions can really mean.
Let me give an example first.
Suppose you keep rolling a dice until you get a 6. Then on average you need to roll the dice 6 times. Although you say that the expected (meaning: average) number of rolls is 6, you shouldn't plan to get the first 6 in exactly 6th roll. You have some 66.5% chance of getting the first 6 with 6 or fewer rolls. But you have also 33.5% chance of not getting it with the first 6 rolls. If that is the case, after completing the first 6 rolls you still need on average 6 more rolls to get the first 6  by rolling dice 6 times with no 6's, you are no closer to getting a 6 than you were on the very begining. It is possible (with 2,6% chance) to roll a dice 20 times without getting a single 6.
There is a similar phenomena with the AP26 search. The search progress indicates that we should have been expecting on average one AP26 by now. But that doesn't mean that AP26 is going to pop up any day. There is even some 5% chance of so a large misfortune that we will not have a single AP26, when we will be eligible to expect 3 of them. Simply the length of such a search is very unstable and very dependent on good or bad luck.
You have probably observed how unpredictably AP24 and AP25 were showing up. There were long periods with no single AP24, while there were also short periods with a bunch of AP24's found. You can calculate how long, on average, you have to wait for an AP24, but if at some moment you start waiting for a new AP24 to appear, the wait time will often be much longer or much shorter then the average.
As far as expected remaining search time is considered, now we are no closer to find an AP26 than we were at the search start. We have no other choice than just to keep searching.
If I was going to give some predictions for the further search right now, I would say:
* We have some 6070% chance of getting AP26 before the current solution counters double.
* We have some 8590% chance of getting AP26 before the current solutions counters triple.
The bottom line is: this is all about probability of finding AP26 in a given time and one cannot predict the moment when actual solution will appear.
Note that AP25 counter is not a good measure of the search progress as it is due to a large statistical error because of its small value. The nature wasn't kind to us with AP25 as well  on average we should be seeing 6 of them by now. 



I mostly meant "wildly" as hyperbole; I know that the ranges haven't been exceeded by a statistically significant amount.
Thanks, though, for the explanation of how relevant those predictions actually are. That helped clear up a few things. I wasn't sure if there was anything about the AP search itself that allowed for more precision than "x% chance in y amount of time".
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Scott BrownVolunteer moderator Project administrator Volunteer tester Project scientist
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Joined: 17 Oct 05 Posts: 1923 ID: 1178 Credit: 6,444,181,042 RAC: 2,787,121

There is a similar phenomena with the AP26 search. The search progress indicates that we should have been expecting on average one AP26 by now. But that doesn't mean that AP26 is going to pop up any day. There is even some 5% chance of so a large misfortune that we will not have a single AP26, when we will be eligible to expect 3 of them. Simply the length of such a search is very unstable and very dependent on good or bad luck.
The dice example is a nice classical statistics example for the use of the binomial distribution. But with the die roll, we know if it is binomial. With AP26, I am curious what distributional assumption you are using to make the prediction? Given that the predictions have been off with lower AP2024 being observed at a higher than expected rate and AP25 and 26 at a lower rate, it looks like your distributional assumption in the prediction is too "fat" in the tails?
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141941*2^42994381 is prime!




With the last challenge completed we are well beyond any predictions, have nearly doubled them:
20 16513
21 4272
22 1042
23 241
24 59
25 8
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