Fourth probable prime found, one to go!

[mdettweiler]

Originally Posted by henryzz

i am not sure we know enough about prime numbers to work out accurate predictions
on this project the predictions keep on being way too high for instance

First of all, huge congratulations to engracio on an enormous PRP!

Secondly, I was thinking the same thing as henryzz, that possibly there's something we don't yet know that's causing our predictions to be way too high. I wonder if there's something specific to the Dual Sierpinski conjecture that makes it more "prime-dense" than others?

[engracio]

Originally Posted by philmoore

I don't have any problem notifying everyone right away, but I would probably prefer to do it via email or PM than post it on the Forum. I wrote Jeff just a couple of hours after I received your email, telling him to pull all the 41693 numbers from his 5.15-5.16M reservation until we knew for sure, but I also asked him if he would be willing to finish the 5.09-5.10M reservation so that we knew whether your prp was truly the smallest. But if he hadn't wanted to do that, I was willing to finish that set myself. I wrote Ben the day afterwards, telling him the situation and asking if he was willing to finish the 41693 numbers in his 4.97-5.00M reservation, and I also let Kent, who was sieving, know fairly early on. The reason I wanted to finish the smaller numbers is to complete the Sloane sequence A067760 up to (78557-1)/2:
http://www.research.att.com/~njas/sequences/index.html?q=A067760&language=english
But if anyone wants to jettison the rest of their tests next time, no problem, I am willing to finish them. The work stoppage may be more than a day or two if the next prp is quite a bit larger, but the prospect of a false pseudo-prime is pretty remote.

Of course, this assumes that the project stays small. If it grows a great deal, it would probably make more sense to do as you suggest.

Thanks Phil that makes sense to me, just to make sure that we have all the blocks checked. I did not realize that was your reasoning. I believe I completed all other wu prior to the 4th PRP, if not it would be just a couple.

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[engracio]

Originally Posted by Jeff Gilchrist

Congrats everyone, great work!

Now I'm starting to wonder if paleseptember = engracio as a second account because how is it possible for anyone besides him to find a PRP in this project?!?!?!

Just kidding of course, looks like I just missed that reservation. Maybe next time.

I hear you man. Unless I am mistaken he is down there and we is up here, so no we be not cousins.

On the other hand, to up our chances for the last sequence, maybe we need for mister prime of FOB to have him help us with the last one. Last one is the charm to paraphrase a common saying, yea right....

[philmoore]

Mini-Geek calls it the butterfly effect: because he took a single work file 4.80M, that caused this latest prime to be discovered by Engracio rather than Jeff. Jeff, you have my permission to find the next one, but let me say in advance, I will be pleased no matter who finds it!

These primes certainly seem to be consistently showing up early - I just hope this trend continues a little longer. Maybe mdettweiler and henryzz are right, that there is something we don't know that is making the problem a little easier than predicted.

[engracio]

Originally Posted by philmoore

These primes certainly seem to be consistently showing up early - I just hope this trend continues a little longer. Maybe mdettweiler and henryzz are right, that there is something we don't know that is making the problem a little easier than predicted.

Sounds like someone with a little bit of time and lots of know how, they can bite into and be busy for a while.

[geoff]

Congratulations engracio! What a breakthrough for this project.

[Kman1293]

Congrats engracio!

[Zuzu]

Congrats to Engracio and all the group, we're lucky indeed !!!
Regarding the stats for the last prime: The 63M figure for 50% prob looks correct, but I don't understand why "We shouldn't expect 1 prime until 191.63M" according to Mini-Geek...
BTW much congrats and luck, I didn't expect a PRP so early, the 50% prob was around 17M indeed...
(sorry for being spartan in the form but I'm abroad, having not much time...)

[Mini-Geek]

Originally Posted by Zuzu

Congrats to Engracio and all the group, we're lucky indeed !!!
Regarding the stats for the last prime: The 63M figure for 50% prob looks correct, but I don't understand why "We shouldn't expect 1 prime until 191.63M" according to Mini-Geek...
BTW much congrats and luck, I didn't expect a PRP so early, the 50% prob was around 17M indeed...
(sorry for being spartan in the form but I'm abroad, having not much time...)

See the equation I had for calculating a 50% chance:

Originally Posted by Mini-Geek

63.17M=5.15M*2^(X/.19166) with X such that e^(-X)=0.5

e^(-x)=y is an equation meaning that when you expect x random occurrences, the chances of it not happening are y. We're looking for the chances of it happening (i.e. finding a prime), so we really want 1-e^(-x). (note that I left that out up there because for y=0.5, a 50% chance, they're the same) For the one I showed there, I wanted to know the expected number of primes before we had a 50% chance of finding one. It's about 0.693. The 5.15M*2^(X/.19166) part lets me smooth out calculating between doublings, where I know we expect .19166 primes. This means I can see the size we should expect to get to for any expected number of primes.
By the same idea, 191.63M=5.15M*2^(1/.19166), and 1-e^(-1)=0.632...

You can see this is probably about right by noting that .19166*5 is a little under 1, and if you double 5.15M five times you get 164.8M, which is a little smaller than the number I calculated.

[philmoore]

Our new prime is now listed on the Lifchitz site:
http://www.primenumbers.net/prptop/prptop.php.

[Zuzu]

OK I understand your calculation: 191M is the exponent n for which the probability of finding no PRP between 1 and n is equal to exp(-1) = 0.368, provided that no PRP has been found between 1 and 5.15M (thus neglecting the possibility of finding a PRP by double-checking).
IMO the reference probability is more based on 50% but as you have written, for each p in ]0,1[ one can similarily calculate the exponent n(p) for which the probability of finding no PRP between 1 and n is equal to p: n(p)=5.15M*2^(-ln(p)/0.19166))

For low values of p, n(p) increases dramatically: 7.54M for 0.9; 14.58M for 0.75; 63.17M for 0.5 but 775M for 0.25; 21G for 0.1; 261G for 0.05 and 88T for 0.01. There is a reasonable probability of finding the last PRP before say 50M (even before 10M if lucky as we were) but not finding any before say 1G is not to be ruled out.
OTOH for the SoB project, the mean number of primes per doubling being 1.04, the range of outcomes is less extended for the next prime. Provided that no prime has been found before 17.2M and neglecting double-check, n(p) would be equal to 18.5M for 0.9; 20.8M for 0.75; 27.3M for 0.5; 43.4M for 0.25; 80.1M for 0.1; 127M for 0.05 and 373M for 0.01.