20100303, 01:06  #45  
Sep 2006
The Netherlands
677 Posts 
Quote:
Note if it would work, there is a real fast manner to find prime numbers for Mersenne. Vincent 

20100303, 01:16  #46  
A Sunny Moo
Aug 2007
USA (GMT5)
3·2,083 Posts 
Quote:
Code:
A.3. More details/methods used Pfgw can work with numbers from 2 up to almost 2^79700000 (about 24000000 digits). It can find probable primes with Fermat's method, with bases from 2 to 256. To be more precise: The largest FFT is 4 million elements long, with 19 bits per element. GFN's can be tested upto 24M digits, and generic numbers upto 12M digits. To prove a number prime, other methods need to be used. Only a small percentage of all numbers can be easily proven prime. Name a number N, then you must be able to factor N1 or N+1 to 33.33% to find a proof using PFGW. If N1 is factored deep enough, then Pocklington's test can be applied. If N+1 is factored deep enough, then Morrison's test can be applied. If N^21 is factored deep enough, a combined method can be used. A.3.1 Fermat's method Fermat's method is NOT a proof, but more like a quick indicator that a number might be prime. A.3.2 Pocklington's test This test can be used whenever N1 can be factored to 33.33% of the size of N. (Actually, the factored part must be greater than the cube root of N/1000000). This test is conclusive. A.3.3 Morrison's test This test can be used whenever N+1 can be factored to 33.33% of the size of N. (Actually, the factored part must be greater than the cube root of N/1000000). This test is conclusive. A.3.4 FStrong test This test is used when you use the t option, and your factors don't reach the magic 33.33%. It is a strongprimality test, and gives more certainty than a Fermat test, but still is NOT a proof! Last fiddled with by mdettweiler on 20100303 at 01:17 

20100303, 01:36  #47  
Feb 2005
2^{2}·3^{2}·7 Posts 
Quote:
Last fiddled with by maxal on 20100303 at 01:39 

20100303, 02:01  #48 
A Sunny Moo
Aug 2007
USA (GMT5)
3×2,083 Posts 
No, I'm quite sure it isn't a MillerRabin test; I'm pretty sure it's a special type of method (not sure what exactly it isall I know is that PFGW calls it "FStrong test") that arises from an N1 or N+1 test without sufficient factorization. Someone with more familiarity with such things can feel free to elaborate here.
Last fiddled with by mdettweiler on 20100303 at 02:01 
20100303, 13:41  #49 
"Phil"
Sep 2002
Tracktown, U.S.A.
45D_{16} Posts 
I'm guessing it is a strong Frobenius test. Can anyone confirm this?

20100303, 14:37  #50  
Mar 2010
3 Posts 
Quote:


20100303, 16:48  #51  
Aug 2006
1011011101111_{2} Posts 
Quote:
I mean, in a region that large there's a decent chance of a Wagstaff prime popping up anyway  I estimate a 57% chance based on an offthecuff adaptation of Wagstaff's heuristic for Mersenne numbers (this is probably low, since I haven't examined the numbers mod 8). But why do you expect pairs after large gaps? I would expect them to be (after transforming to normalize the probabilities) Poissondistributed. 

20100303, 22:39  #52 
Bemusing Prompter
"Danny"
Dec 2002
California
2^{2}×3×191 Posts 
Tony, I did not know your wife had passed away until just now. I'm really sorry to hear that.

20100304, 06:50  #53 
Feb 2004
France
1617_{8} Posts 

20100304, 22:36  #54 
Mar 2010
3 Posts 
I chose p<5500000 since it is one average gap higher than 4000000 in base 3 with G=.47 instead of 0.56145948. Since the distribution of primes is random (Poisson), the location of the next prime is not deterministic, but 2 large gaps, back to back is very rare. To quote from Caldwell's prime pages
http://primes.utm.edu/notes/faq/NextMersenne.html Noll's nebulous "Island Theory" for Mersennes (which I have never seen quantified) seems to roughly state that Mersennes occur in clumps with gaps between them. Perhaps so, but that is exactly what you would expect from any Poisson process! It seems likely that after large gaps, we have encountered a "clump" of 2 or 3 primes. Even though I know that the gaps are independent, they can not all be uniformly spaced either. 
20100305, 13:00  #55 
Mar 2010
3 Posts 
Sorry, base 2 not base 3

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