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Generalized Fermat Prime Search :
New PRP at n=19
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According to this http://www.primegrid.com/stats_genefer.php, there's a new PRP.
It should enter (or get close to) top 10 on the largest primes list.
Congratulations to the finder and to Primegrid.
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676754^262144+1 is prime  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 14036 ID: 53948 Credit: 475,998,733 RAC: 222,579

Since Genefer is "only" a probable prime test, that number still needs to be checked for primality using PFGW. However, the odds of Genefer falsely reporting it as prime are astronomically small, so this is essentially a necessary formality.
This is really big news. Assuming it checks out, it's the largest prime number ever found at PrimeGrid and the 11th largest prime overall.
Details to follow...
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My lucky number is 75898^{524288}+1  


Mighty exciting!
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Twitter: IainBethune
Proud member of team "Aggie The Pew". Go Aggie!
3073428256125*2^12900001 is Prime!  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 14036 ID: 53948 Credit: 475,998,733 RAC: 222,579

Mighty exciting!
Not nearly as exciting as your new "find", however! :)
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My lucky number is 75898^{524288}+1  


Hopefully the challenge will churn up another :)
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Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 14036 ID: 53948 Credit: 475,998,733 RAC: 222,579

Hopefully the challenge will churn up another :)
Or two. :)
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My lucky number is 75898^{524288}+1  


Hopefully the challenge will churn up another :)
Or two. :)
Two PRPs and the challenge did not start yet!
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676754^262144+1 is prime  


Bleeding heck, that's some good news! :)
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PrimeGrid Challenge Overall standings  Last update: From Pi to Paddy (2016)
 

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 14036 ID: 53948 Credit: 475,998,733 RAC: 222,579

Hopefully the challenge will churn up another :)
Or two. :)
Two PRPs and the challenge did not start yet!
WOW!
When I wrote "Or two" I meant "let's FIND 2 more during the challenge", not that we had just found two.
This is astounding!!!!!!
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My lucky number is 75898^{524288}+1  


Since Genefer is "only" a probable prime test, that number still needs to be checked for primality using PFGW. However, the odds of Genefer falsely reporting it as prime are astronomically small, so this is essentially a necessary formality.
This is really big news. Assuming it checks out, it's the largest prime number ever found at PrimeGrid and the 11th largest prime overall.
Details to follow...
Just a hint: I think that the prime on my signature was checked with PFGW and LLR and then submitted to the primes page with the latter as proving app. LLR was quite a bit faster than pfgw at the time (llravx vs pfgw non avx, if I remember), so John used both for primality check, but submitted the prime with LLR, in order to allow faster checks in the future. Se this message: http://www.primegrid.com/forum_thread.php?id=4065&nowrap=true#49530
I don't know which app is faster now, specially with such huge primes, but if LLR is still faster, you could use it instead of PFGW.
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676754^262144+1 is prime  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 14036 ID: 53948 Credit: 475,998,733 RAC: 222,579

I don't know which app is faster now, specially with such huge primes, but if LLR is still faster, you could use it instead of PFGW.
LLR is being used, if for no other reason than because it checkpoints and PFGW does not.
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My lucky number is 75898^{524288}+1  


I'm confused. Is there a second GFP that was found just prior to the challenge or not?
341112^524288+1
This is the only prime I've seen so far.  


Yes there is, but the person to first hand in the result did have auto reporting switched on and so far they have been unable to get a hold of him(/her) to get their real life name. Therefore they haven't been able to report it yet and no official announcement has been made.
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PrimeGrid Challenge Overall standings  Last update: From Pi to Paddy (2016)
 


Is this second GFP larger or smaller than 341112^524288+1?  

Scott BrownVolunteer moderator Project administrator Volunteer tester Project scientist
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Joined: 17 Oct 05 Posts: 2416 ID: 1178 Credit: 19,976,307,436 RAC: 19,272,376

Larger.
 


Looks like there's a new PRP at n=19. It's the 3rd on this range since the project has been ported to BOINC.
Congratulations to the finder, who ever he or she may be.
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676754^262144+1 is prime  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 14036 ID: 53948 Credit: 475,998,733 RAC: 222,579

Looks like there's a new PRP at n=19. It's the 3rd on this range since the project has been ported to BOINC.
Yes, another record GFN prime has been found. It's still a PRP and needs to be run through a longer proof test, which will take a few days. Once it's confirmed, and notifications, etc., are done, it will be the new 11th largest known prime number, largest known GFN prime, and largest prime ever discovered at PrimeGrid.
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My lucky number is 75898^{524288}+1  


475856^524288 + 1 has been added to Top5000...
http://primes.utm.edu/primes/page.php?id=108818
Is that the one mentioned before?
Congratz to the finder!  


Well congratulations to the finder on another Mega (and record) GFN. What is really amazing is that according to Yves Gallot's method for calculating the expected number of GFNs in a range, we would only have expected 1 GFN for N=524288, and instead we have found 4  which has about a 1.5% probability.
So either we have just been exceedingly lucky with N=524288, or there are in fact more GFNs at high n than predicted, which would be exciting indeed for the GFNWR project!
____________
Twitter: IainBethune
Proud member of team "Aggie The Pew". Go Aggie!
3073428256125*2^12900001 is Prime!  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 14036 ID: 53948 Credit: 475,998,733 RAC: 222,579

What is really amazing is that according to Yves Gallot's method for calculating the expected number of GFNs in a range, we would only have expected 1 GFN for N=524288, and instead we have found 4  which has about a 1.5% probability.
The theory behind the math for those predictions is suspect. It's essentially, "This formula models what we've seen in the past, therefore it should be a reliable prediction of what we'll see in the future."  without giving any justification for why such an extrapolation would be valid.
I'm not surprised we found more. I am a little surprised (although I shouldn't be because I do know better) that we found three so close together, but that's actually fairly typical of prime numbers (as well as pseudo random numbers that they resemble).
It will be REALLY interesting to see where the first WR GFN pops up. By the existing formula, we shouldn't find any, but the formula stopped being an accurate predictor quite a while ago. It underestimated the primes found at n=18 also. In fact, if you take a look at the primes shown on Yves website, the formula erred low on everything beyond that for which he had data. In essence, the formula worked fine for distributions that were already known. As for predicting future primes, not so much.
Then again, that's the nature of science and mathematics. Use existing theory to create a hypothesis, test the hypothesis, and adjust (or discard) the hypothesis based on the experimental results.
It's time for a new formula. :)
Of course, the really fun part of all this is that we're in completely uncharted territory and nobody really can tell you when or if that 13 or 14 million digit prime number is going to pop up. This is exciting.
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My lucky number is 75898^{524288}+1  

Yves GallotVolunteer developer Project scientist Send message
Joined: 19 Aug 12 Posts: 842 ID: 164101 Credit: 306,514,171 RAC: 5,382

Hi all, first of all congratulations on your search.
GPU computing is a new step for the prime number search, and you are the avantgarde!
A lucky man wrote that “The theory behind the math for those predictions is suspect” but my guess is B = 3,380,000 :)
I think that any B in [2012 ; 10^7] is a good choice.
Note that C_n ~ n log 2 and Li(B) ~ B / log B. Then B = 2^n is the “generic” estimate.
Here, B is smaller because C_22 = 17.3 and 22 log 2 = 15.9.
“It's time for a new formula.”…
On one hand, the results: for B in [2 ; 500000]
n = 16, primes: 9, expected: 7,
n = 17, primes: 4, expected: 3.5,
n = 18, primes: 4, expected: 2,
n = 19, primes: 4, expected: 1.
Another interesting result is with Mersenne primes:
p in [10^7 ; 5.10^7], primes: 9, expected: 4.
Then the number of huge primes is larger than expected!
On the other hand, the expected number of primes is based on the BatemanHorn conjecture (see wikipedia:BatemanHorn_conjecture).
This conjecture is not based on “what we've seen in the past” but is a generalization of some theorems.
If the polynomial is x, the conjecture is the prime number theorem and if it is ax+b, the conjecture is Dirichlet's theorem.
It is practical. C is linked to the sieve process: it is the ratio of the numbers that pass through the sieve.
The conjecture is just the mathematical form of: the probability of a number n = f(x), which has no small factor, being prime depends on its size but not on f.
Then, what? We are lucky, may be. Or an unknown property was found, that would be very interesting!!!
The results of sieving are a first indicator. What are they for n =18, 19 and 22?
 

JimBHonorary cruncher Send message
Joined: 4 Aug 11 Posts: 920 ID: 107307 Credit: 989,312,040 RAC: 1,991

The results of sieving are a first indicator. What are they for n =18, 19 and 22?
As of August 20, 2012 at 1100 UTC:
n=18 is sieved to 2510P with 17 300 322 candidates remaining in the sieve
n=19 is sieved to 19100P (gap from 18446P19000P) with 16 546 522 candidates remaining in the sieve
n=20 is sieved to 19900P (255P in gaps) with 18 347 008 candidates remaining in the sieve
n=21 is sieved to 20000P (750P in gaps) with 20 396 612 candidates remaining in the sieve
n=22 is sieved to 22400P (1200P in gaps) with 21 963 320 candidates remaining in the sieve
In all cases, these numbers represent the results of sieving only and don't take into account any candidates already primetested by various flavors of Genefer.  


Jim,
Those are the candidates within the full range sieved. Could you please post the number of candidates bellow 500000 (used by Yves a couple of posts ago)?  

JimBHonorary cruncher Send message
Joined: 4 Aug 11 Posts: 920 ID: 107307 Credit: 989,312,040 RAC: 1,991

As of 21 August 2012 at 0030 UTC:
Candidates Candidates
n limit gaps <=500K Total
18 2510P 0 86 607 17 300 322
19 19100P 554P 82 895 16 546 522
20 19900P 55P 91 242 18 342 741
21 20000P 650P 102 316 20 378 158
22 22400P 1204P 110 064 21 963 320
As previously, these counts represent candidates removed by sieving only.  

Yves GallotVolunteer developer Project scientist Send message
Joined: 19 Aug 12 Posts: 842 ID: 164101 Credit: 306,514,171 RAC: 5,382

The basic property for the estimate of the number of GF primes is:
let p = k*2^(n + 1) + 1 be prime. Then it divides 2^n/p GF numbers of the form b^2^n + 1.
If we sieve R odd numbers, for all p < p_max, the expected candidates that left are
prod_{p < p_max} (1  2^n/p) * R.
By Mertens' 3rd theorem:
prod_{p < p_max} (1  2^n/p) = 2 * C_n / (e^gamma * log(p_max)),
where gamma is Euler's constant.
If we sieve [2, B_max],
Candidates / (B_max/2) = 2 * C_n / (e^gamma * log((p_max))
and finally we have
Candidates = e^gamma * C_n * B_max / log((p_max).
We obtain for B_max = 500000:
n C_n p_max ratio Expected candidates
18 13.0 2.510E+18 0.345 86140
19 13.1 1.855E+19 0.332 82890
20 14.5 1.985E+19 0.366 91609
21 16.1 1.935E+19 0.407 101775
22 17.4 2.120E+19 0.439 109768
and B_max = 1E8:
n C_n p_max ratio B_max Expected candidates
18 13.0 2.510E+18 0.345 17228044
19 13.1 1.855E+19 0.332 16577985
20 14.5 1.985E+19 0.366 18321722
21 16.1 1.935E+19 0.407 20355000
22 17.4 2.120E+19 0.439 21953527
The statistics of sieving is close to estimate :)
 

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New PRP at n=19 