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I know the question has been asked before but i wasn't sure if anyone was ever able to figure it out. Does anyone know what the probability of a certain N value turning out to be prime, and where is the current 50% line for when we'll find the next prime. I'm just trying ot figure out if this is a DC project I can hope to see finish in my life time.
The probability that any exponent is a prime is the Proth Weight for the k value divided by ln(x). This is the prime number theorem weight adjusted for the unusual divisibility properties of Proth numbers - things like none are divisibly by 2 and 1/2 are divisibly by 3 instead of 1/2 and 1/3. Yves Gallot has Proth weights in Table 5 of this paper.Quote:
Originally posted by Keroberts1
I know the question has been asked before but i wasn't sure if anyone was ever able to figure it out. Does anyone know what the probability of a certain N value turning out to be prime, and where is the current 50% line for when we'll find the next prime. I'm just trying ot figure out if this is a DC project I can hope to see finish in my life time.
Adding up all the weights for all the exponents, there is a 50% chance of at least one prime between 4.2M and 5.4M.
So if we put the numbers in order of probability...or weight...it would look like this:
K Weight Status
44131 0.31 Prime Found
54767 0.28 Prime Found
55459 0.25 Still Going
5359 0.25 Still Going
10223 0.23 Still Going
4847 0.20 Still Going
24737 0.20 Still Going
21181 0.19 Still Going
33661 0.19 Still Going
69109 0.18 Prime Found
27653 0.12 Still Going
28433 0.11 Still Going
46157 0.11 Prime Found
65567 0.09 Prime Found
19249 0.08 Still Going
22699 0.07 Still Going
67607 0.07 Still Going
So the first two predictably fell...by weight it's looking good for 55459 and 5359 since so many of the PRP test are going toward them...Think of the work reduction if we could find them Prime soon. That's ~700 pending test just between those two numbers alone. I see we probably got lucky with the other three Primes.
I must be doing something wrong, I got that for a test on 55459 we should have a .01636% chance of getting a prime adn then that would mean one out of 6109 in the chance of getting a prime but we've already tested over 25000. Have we just been extremely unlucky?
I'm getting similar numbers or many of the others too.
We are presently testing in the vicinity of n=4.2*10^6. Thus
x=k*2^(4.2*10^6)+1
ln(x) = ln(k)+4.2*10^6*ln(2)
If we had done no trial factoring nor P-1 factoring, the probability an individual test would be prime would be 8.5*10^-8. SoB started with the k value at about 540,000 - the probability at that time was about 6.7*10^7. Roll up the probability for all the numbers between n=540,000 and n=4.2M, and the probability of finding at least one prime in that interval is about 52%.
Since nobody replied to this: I expect to see the 6th prime in SoB, and (a lot later) the 7th. I doubt I'll stick around after that, as there seems to be a back-of-the envelope expectation of hundreds of years before the 8th is found (given the current computational strategies).Quote:
Originally posted by Keroberts1
I'm just trying ot figure out if this is a DC project I can hope to see finish in my life time.
Personally I think we are overdue on 55459 and 5359. I think we should be about to the point of where 10223 should fall. If my basic math skills are correct..if these three should fall to Primes then out of the 521K+ test still needing to be done on n<20M would drop by over 260K test...so basically half the work would suddenly be taken out of the project. 1/2 of our resources are tied up in finding primes for just 3 of the 12 remaining. Once these three fall..or even one of them fall...with the added membership boost that a project brings when things start going gravy...and the freed up resources....a domino effect could take place and speed this entire project up.
I for one see it as a great project that can easily be finished well within my lifetime..I'm just 34...I even see the completion of RC5-72 in my lifetime..and that is currently projected at +1500 years...but CPU speed doubles every couple of years...everyone is getting more than one computer at home...these projects will be crushed in due order.
You possibly can't tell which of the numbers will fall next.Quote:
Personally I think we are overdue on 55459 and 5359. I think we should be about to the point of where 10223 should fall.
no, but you can say which ones are most likely to fall next
These numbvers help alot and give a good puicture of how many tests are left to be performed before the project will be solved but how does the sieve depth change the chances the number will come back prime. I know the sieve isn't gonna find a prime and isn't going to many any particular test mor likely to be rime I'm just saying if many factors are foudn then less test will have to be erformed therefore the average chance that a particular test will come out prime is improved by this. I just wonder how much. If anyone knows how ot calculate these values lemme know.
Thanks
The Proth-Weight estimates can be used to find the probability some exponent in a range is prime. This calculation is theoretically the number of candidates left after you remove those divisible by 3 and 5 and 7 etc. But this the same thing that sieving does. So what sieving does is it reduces the number of tests required to completely test the range without changing the probability there is a prime in the range.Quote:
Originally posted by Keroberts1
These numbvers help alot and give a good puicture of how many tests are left to be performed before the project will be solved but how does the sieve depth change the chances the number will come back prime.
For example, if you calculated there was a 1 in hundred chance of a prime in a range of 100,000, then if you were to just test each exponent, there would be a 1 on 10,000,000 that each exponent was a prime. But if sieving had reduced that range to 500 numbers, then each of THOSE 500 tests would have one chance in 50,000 of being prime. If the range was further sieved so that only 100 candidates were left, the each of THOSE 100 tests would have one chance in 10,000 of being prime.
ok where ca ni find info on the number of factors found in each range
ok where ca ni find info on the number of factors found in each range. I think I may have see nit here somewhere before, but I've forgotten where.