20110209, 06:41  #1 
Jun 2010
Pennsylvania
13·71 Posts 
'Verifying' a Mersenne prime vs. 'proving' it
Hi,
I'm not sure which subforum is the best place to ask this, but I have a question about the GIMPS procedures for confirming a Mersenne prime. I read here (http://primes.utm.edu/mersenne/index.html#known) that 20996011 was discovered in 2003, and the associated link (http://primes.utm.edu/notes/20996011/index.html) informs the reader that the new prime was "verified" by two other folks back then. So far, so good: I understand the process. But then there is this page (http://mersenne.org/report_milestones/) which reports that 20996011 wasn't "proven" to be the 40th Mersenne prime until July 2010  that is, close to seven years after it was "verified." What's the difference between "verifying" and "proving" a Mersenne prime? I can understand each process separately (the wavefront for LLD is far behind that for LL, and one would like to confirm a new MP ASAP), but not how the two concepts relate to each other. If a MP isn't "proven," has it really been "verified"? Or, if it has been "verified," why does it need to be "proven"? Thanks in advance for elucidating these concepts for me. Rodrigo Last fiddled with by Rodrigo on 20110209 at 06:41 
20110209, 06:51  #2 
Dec 2007
Cleves, Germany
529_{10} Posts 
You verify that a number is prime, and prove its place in the sequence of Mersenne primes (by not verifying any previously unknown lower ones, so to speak).

20110209, 07:08  #3  
A Sunny Moo
Aug 2007
USA (GMT5)
3×2,083 Posts 
Quote:
Note that for other, smaller (nonMersenne) primes, the standard of what's considered "proof" for a prime is not often so stringent. In such cases, two separate tests with the same program on different computers (usually both x86based) that both agree are considered sufficient proof for the number to be accepted on the List of the 5000 Largest Known Primes. (At least one test is assumed to have been done by the person submitting the prime, and a second test is done by the top5000 site's internal verification machines.) For Mersenne primes, due to their immense size and the correspondingly long tests involved (i.e. greater chance of a hardware error having occurred during the test), additional tests are routinely performed by nonPrime95 software on nonx86 hardware to eliminate the chance of a false positive due to a Prime95 bug or a glitch in the x86 microarchitecture in general. Both are rather unlikely, but because of the large amount of attention given to Mersenne primes (as they have traditionally been tested to sizes much higher than other forms), it's good to make "extra sure". That way you don't have to issue an embarrassing retraction of the proclaimed momentous discovery. 

20110209, 17:35  #4 
Jun 2010
Pennsylvania
13·71 Posts 
@ckdo
@mdettweiler Thanks a bunch for explaining! I think I get it now. First somebody finds a (possible) Mersenne prime, and soon thereafter George verifies that it's "a" Mersenne prime. Then, years later, doublechecking establishes definitively that it's "the" Xth Mersenne prime. Right? Rodrigo 
20110209, 17:42  #5 
A Sunny Moo
Aug 2007
USA (GMT5)
3×2,083 Posts 
Yep! One quick note, though: it's generally not George himself who does the verification of prospective primes. He does a quick run of the last few iterations from the save file which Prime95 will save in case of a prime, and which he'll ask the finder to email him right away; but the full verifications are done by a couple of supercomputers that some members of this forum have access to. They usually manage to get the verification done in a week or so, which is pretty quick considering that the original test at leadingedgefirstpassLL size generally would take about a month, and that the supercomputers are at somewhat of a disadvantage since Prime95 is a lot better optimized than all the nonx86 LL testing programs currently out there.
Next time a Mersenne prime is found, there will probably be at least one or two people running the verification on a GPU with CUDALucas (possibly also with the forthcoming GPULucas if it's out by thenas with CPUs, the more different programs you can test it with, the better to reduce the probability of software bugs). A modern GPU like a GTX 460 or GTX 580 is able to test a leadingedge number at least as quickly as one of the aforementioned supercomputers. Last fiddled with by mdettweiler on 20110209 at 17:59 
20110209, 18:58  #6 
Jun 2003
1219_{16} Posts 

20110209, 19:00  #7 
Jun 2010
Pennsylvania
39B_{16} Posts 
mdettweiler,
Thanks for elaborating on the process, it's very much clearer in my head now. It IS amazing that a modern GPU could compete with a supercomputer on something like this! Rodrigo 
20110209, 19:12  #8 
A Sunny Moo
Aug 2007
USA (GMT5)
6249_{10} Posts 

20110209, 22:17  #9  
Nov 2003
2^{5}×233 Posts 
Quote:
40th such prime until 2010 when all other possible candidates had been eliminated. 

20110210, 00:31  #10 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2·4,111 Posts 
Sounds like an entry in the Wiki is needed.

20110210, 03:13  #11  
Jun 2010
Pennsylvania
13·71 Posts 
Quote:
It was when I honed in on the key word "known" that it really clicked: That was the fortieth time that a Mersenne prime was ever discovered, but as to whether it was Number 40 in the sequence of positive integers defined as Mersenne primes  that's what didn't get proven till seven years later. Hopefully, this thread will help to make this aspect of the project clearer for at least a few people out there. Rodrigo 

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