M47
M47 | |
---|---|
Prime class : | |
Type : | Mersenne prime |
Formula : | M_{n} = 2^{n} - 1 |
Prime data : | |
Rank : | 47 |
n-value : | 43,112,609 |
Number : | 316470269330...166697152511 |
Digits : | 12,978,189 |
Perfect number : | 2^{43,112,608} • (2^{43,112,609}-1) |
Digits : | 25,957,378 |
Discovery data : | |
Date of Discovery : | 2008-08-23 |
Discoverer : | Edson Smith |
Found with : | Lucas-Lehmer test Prime95 on Dell Optiplex 745 |
Credits : | George Woltman Scott Kurowski et al. (GIMPS & PrimeNet) |
M47 normally refers to the 47th Mersenne prime 2^{43,112,609}-1, in order of size from the smallest to greatest. This is the primary usage and what is refered to in the rest of this article. For clarification about other possible usages refer to the Nomenclature and notation article. The number now refered to as M47 was actually the 45th Mersenne prime found. M45, M46, and M47 were discovered in the order of M47, M45 (2 weeks later), then M46 (8 months later).
On 2018-04-08 all tests below 2^{43,112,609}-1 were verified by GIMPS, officially making it the 47th Mersenne prime.
Contents
Discovery
On 2008-08-23 a Dell Optiplex 745 computer, running Prime95, completed the Lucas-Lehmer test for the number 2^{43,112,609}-1 and found out that it was a prime number. The computer was stationed at a lab in the Mathematics Department at University of California, Los Angeles.
Upon discovery, an audio alert was activated and an automatic notification was sent to PrimeNet. PrimeNet sent an automatic e-mail to George Woltman. He then contacted Edson Smith, who lead the team at UCLA.
This was the first prime found larger than ten million digits, it has 12,978,189 decimal digits.
Verification
To confirm that there were no errors in the hardware (such as the Pentium Bug or a random bit error or software, the number had to be verified by running the Lucas-Lehmer test on at least one different machine (not just a separate physical computer, but one based upon a different CPU architecture) and using a different program.
As a very quick check against errors, George Woltman obtained a "save file" from Edson Smith and re-ran the last few thousand iterations on one of his machines. No problems were detected. He also started testing the exponent on one of his machines. (Even though it used a similar processor and the same software, this step was useful for finding potential errors before the test was complete.) George notified a select group of volunteers that have access to much faster computers. They started verification runs using these different types of computers and different software. They are:
- Tony Reix (France) of Bull S.A. running Glucas (written by Guillermo Ballester) on 16 1.6GHz Itanium2 CPUs of a Bull NovaScale 6160 HPC server in Grenoble
- Jeff Gilchrist (Canada) ran Glucas on up to 16 1.6 GHz Itanium2 CPUs of a server at SHARCNET in Ottawa.
- Tom Duell (USA) and Rob Giltrap (New Zealand), both of Sun Microsystems, using Mlucas doing 2 separate runs:
- The machines were in Menlo Park, California, USA. These were the first verifications completed and took 13 days.
Throughout the verification runs, the residue values were exchanged amongst the various individuals. If they did not match, an error would have been indicated. This is were George's run with Prime95 (although slower) was useful. If his check had shown up different than the others, it would have indicated a problem with either his software or hardware.
Prize
This was the first prime number, of any sort, found larger than 10,000,000 digits. The Electronic Frontier Foundation had an established $100,000 prize for the first prime greater than 10,000,000 digits. GIMPS claimed the prize and divide it with $50,000 going to UCLA, $25,000 going to a charity, and the remainder being used by GIMPS to provide awards to other finders of Mersenne primes and to cover expenses.