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Mersenne Prime:     more detail

Some notes on multiplicative congruential random number generators with Mersenne prime modulus [2.sup.61]-1.: An article from: Journal of the South Carolina Academy of Science by James Harris, 2003-09-22 Illinois. University. Digital Computer Laboratory. Report by Donald Bruce Gillies, 1964 Mersenne's numbers by Raymond Clare Archibald, 1935

lists with details

1. Mersenne Prime Search
   An organized search for mersenne prime numbers. Free software provided.Category Science Math Number Theory Prime Numbers Mersenne M6972593 is the 38th mersenne prime. GIMPS has finished testing and doublecheckingall Mersenne numbers below M6972593. 39th Known mersenne prime Found!!  
   http://www.mersenne.org/prime.htm
   
   Extractions: NEW! Version 22! GIMPS Forums! Pages available in Danish Dutch French German ... Polish , and Spanish . Warning: These translations may not be up-to-date - use AltaVista as necessary. 62722^131072 + 1 prime. At 628,808 digits this is the 5th largest known prime number. This moves GIMPS first prime discovery, M1398269, into 6th place. Now I must change the banner of every mersenne.org web page! Congratulations to discoverer Michael Angel and to Yves Gallot who wrote the program that discovered this generalized Fermat prime. M6972593 is the 38th Mersenne prime. GIMPS has finished testing and double-checking all Mersenne numbers below M6972593. This proves there are no smaller undiscovered Mersenne primes. Version 22 released. Although not a required upgrade, this new version has many small improvements over version 21. The download page has the full details. GIMPS forums. Here you can chat with fellow GIMPS members, get help with installation questions, learn more about how GIMPS works, etc. New Primality Testing Algorithm.

2. Mersenne Prime Mailing List FAQ
   The mersenne prime Mailing list FAQ By Lucas Wiman, and a Cast of Several Please send comments or suggestions to lrwiman  
   http://www.tasam.com/~lrwiman/FAQ-mers
   
   Extractions: < B1). The actual assumption is somewhat more strict than this, but it is required that P-1 be B1 smooth. Say that we have a prime integer p. By Fermat's little theorem, a^(p-1)==1 mod p, a^(p-1)-1==0 mod p. That is to say, p divides (a^(p-1)-1). Now, what if we find a^((p-1)*q) mod p? Well, that is still ==1 mod p (since (a^b)^c==(a^b mod d)^c mod d). So, p still divides a^((p-1)*q)-1. Let p divide n. Assume that p-1 is B1 smooth (this is more like hoping that p-1 is B1 smooth, since we don't know what p is). Then we construct the smallest number q, such that if d < 2^n-1 <= 10^d-1 10^(d-1) < 2^n <= 10^d log_2(10^(d-1)) < n <= log_2(10^d) (d-1)*log_2(10) < n <= d*log_2(10) d-1 < n/log_2(10)

3. The Largest Known Primes
   where p is prime. mersenne prime. A prime number of the form  
   http://www.utm.edu/research/primes/largest.html
   
   Extractions: largest twin ... Mersenne , and Sophie Germain The Complete List of the Largest Known Primes Other Sources of Prime Information Euclid's Proof of the Infinitude of Primes Primes: Home Largest Proving How Many? ... Mailing List Note: The correct URL for this page is http://www.utm.edu/research/primes/largest.html An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. For example, the prime divisors of 10 are 2 and 5; and the first six primes are 2, 3, 5, 7, 11 and 13. ( The first 10,000 , and other lists are available). The Fundamental Theorem of Arithmetic shows that the primes are the building blocks of the positive integers: every positive integer is a product of prime numbers in one and only one way, except for the order of the factors. (This is the key to their importance: the prime factors of an integer determines its properties.) The ancient Greeks proved (ca 300 BC) that there were infinitely many primes and that they were irregularly spaced (there can be arbitrarily large gaps between successive primes ). On the other hand, in the nineteenth century it was shown that the number of primes less than or equal to

4. Mersenne Prime Freeware
   Freeware to search for mersenne primes. Pointers to FreeLIP, GMP, HFloat, apfloat, CALC.  
   http://www.samurai.com/mersenne/freeware.html
   
   Extractions: Mersenne Prime Freeware "Share and Enjoy" The Amdahl 6 's motto Mirror sites For increased performance, this page has been mirrored. Please choose the site closest to you for the best performance. www2.netdoor.com Conrad Curry, Southern US, master site www.samurai.com Bryan Fullerton, Canadian mirror Software Go to the GIMPS homepage . George Woltman's programs are the fastest on Intel chips running Windows 3.x, Windows 95/98/NT, OS/2, DOS, Linux and UNIXWARE. Check out Ernst Mayer's source code timings page for guidance as to which code is best for their platform. This is Dr. Richard Crandall's original Mersenne number testing program. Written in generic 'C'. George Woltman's lightening fast Pentium program has its roots in this code. Reported to be OK mers 6.50 README HTML file about mers package mers.tar.gz binary 'tar xzf mers.tar' (FSF/GNU tar) or 'gunzip mers.tar; tar xf mers.tar'

5. Mersenne Primes: History, Theorems And Lists
   Contents include some historical notes, discussions about perfect numbers and different theorems, and a table of known mersenne primes. See also Where is the next larger mersenne prime? and Mersenne heuristics  
   http://www.utm.edu/research/primes/mersenne.shtml
   
   Extractions: History, Theorems and Lists Early History Perfect Numbers and a Few Theorems Table of Known Mersenne Primes The Lucas-Lehmer Test and Recent History ... Conjectures and Unsolved Problems See also Where is the next larger Mersenne prime? and Mersenne heuristics For remote pages on Mersennes see the Prime Links' Mersenne directory Primes: Home Largest Proving How Many? ... Mailing List Many early writers felt that the numbers of the form 2 n -1 were prime for all primes n , but in 1536 Hudalricus Regius showed that 2 -1 = 2047 was not prime (it is 23 89). By 1603 Pietro Cataldi had correctly verified that 2 -1 and 2 -1 were both prime, but then incorrectly stated 2 n -1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29. Sometime later Euler showed Cataldi's assertion about 31 was correct. Enter French monk Marin Mersenne (1588-1648). Mersenne stated in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2 n -1 were prime for n 31, 67, 127 and 257

6. Mersenne Prime Digits
   Landon Curt Noll lists the decimal, English, and American expansions of all the known mersenne primes.Category Science Math Number Theory Prime Numbers Mersenne mersenne prime Digits and Names. NOTE As of $Date 2003/01/03 131333 $ the lastmersenne prime shown above was the largest known prime. Some useful links.  
   http://www.isthe.com/chongo/tech/math/prime/mersenne.html

7. The Mersenne Prime Homepage
   Notes and links by Simon RubinsteinSalzedo.  
   http://www.albanyconsort.com/mersenne/
   
   Extractions: MERSENNE PRMES Nayan Hajratwala, George Woltman , Scott Kurowski, et. al. found the largest Mersenne Prime ever! It is 2^6972593-1. It is believed to be the 38th Mersenne Prime. It was found by a program by the Great Internet Mersenne Prime Search My club, GIMPS , has the program, called Prime, or Prime 95, by George Woltman Visit the mailing list of Mersenne Primes . P. S. If you don't want a lot of e-mail DON'T click here. Yes. I'm a member of that, too. CELEBRATE MERSENNE DAY EVERY DECEMBER 7TH! Mersenne Pages Mersenne Mailing list Luke's Marin Mersenne Page http://users.delta.com/mersenne/prime.txt The Largest Known Prime by Year: A Brief History ... My homepage

8. ComputerUser.com Mersenne Prime Search
   The Great Internet mersenne prime Search (GIMPS) harnesses the power of thousands of small computers like yours to solve  
   http://www.computeruser.com/resources/links/prime
   
   Extractions: What is the Great Internet Mersenne Prime Search? You could discover one of the most coveted finds in all of Mathematics - a new Mersenne prime number. Join in on this fun, yet serious research project. All you need is a personal computer and a lot of luck. The Electronic Frontier Foundation is offering a $100,000 award to the first person to discover a ten million digit prime number! Later this year, we will make available free software where you can find such a prime number along with details on how the award money will be divided. The Great Internet Mersenne Prime Search (GIMPS) harnesses the power of thousands of small computers like yours to solve the seemingly intractable problem of finding HUGE prime numbers. Specifically, GIMPS looks for Mersenne Primes, expressed by the formula 2P-1. Over 8,000 people have contributed computer time to help discover world-record Mersenne primes. Thoughout history, the largest known prime number has usually been a Mersenne prime. Instructions: Step One: Download the free software

9. GIMPS MERSENNE PRIME WORLD RECORD LARGEST FOUND BY GORDON SPENCE
   Marin Mersenne (15881648) was a Franciscan friar who lived most of his life in Parisian cloisters. The concept of a mersenne prime is evolved from that of a perfect number.  
   http://www.rugeley.demon.co.uk/gimps/prime.htm
   
   Extractions: Number Theory Topic On Sunday 24 th August 1997 I found the 36 th Mersenne Prime at the time it was the world's largest known prime number. It took my Pentium 100 PC fifteen days to prove the number prime. David Slowinski has verified the result on a Cray T90 Supercomputer. This new number is digits in length. George Woltman's announcement Chris Caldwell's announcement Read about it in Science News Article in the London Times ... CNN Article Please note though that I was just the lucky one of over 4500 volunteers all searching for these gigantic numbers, why not join us...... Stands for the Great Internet Mersenne Prime Search, and is a world wide project involving over 4500 participants (@ September 1998), coordinated by George Woltman who maintains the GIMPS site and Scott Kurowski who runs the Primenet Internet Server . Every member of the group runs some Free Software Please note that this size of task would have previously only been possible on large Supercomputers such as the Cray that was used to verify this result, but by running the software on thousands of individual machines we can collectively surpass the power of even the most powerful supercomputer.

10. BBC News | SCI/TECH | Number Takes Prime Position
    For example, the number 10 is not prime because it is divisible by 2 and 5 A Mersenneprime is a prime of the form 2 P 1 The first mersenne primes are 3, 7, 31  
    http://news.bbc.co.uk/hi/english/sci/tech/newsid_1693000/1693364.stm
    
    Extractions: George Woltman, Gimps founder The new number, expressed as 2 -1, contains 4,053,946 digits and would take the best part of three weeks to write out longhand. The prime number - a number that can only be divided by one and itself - was discovered by Michael Cameron, a 20-year-old Canadian participant in a mass computer project known as the Great Internet Mersenne Prime Search (Gimps). Mersenne primes are important for the theory of numbers and they may help in developing unbreakable codes and message encryptions. The Gimps project spent 13,000 years of computer time to find the new prime number. Big effort Cameron used an 800 MHz AMD T-Bird PC, running part-time for 45 days to find the number.

11. Prime Numbers
    Largest known mersenne primes.  
    http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/node10.html

12. 39th Mersenne Prime Discovered
    6, 2001 Michael Cameron, a 20 yearold volunteer in a worldwide research projectcalled the Great Internet mersenne prime Search (GIMPS), has discovered the  
    http://www.mersenne.org/13466917.htm
    
    Extractions: -1 is now the Largest Known Prime. SAN DIEGO, California and ORLANDO, Florida, December 6, 2001 Michael Cameron, a 20 year-old volunteer in a worldwide research project called the Great Internet Mersenne Prime Search (GIMPS) , has discovered the largest known prime number using his PC and software by George Woltman and Entropia, Inc. as part of an international grid of more than 205,000 interconnected computers operated by the company. The new number, expressed in shorthand as 2 -1, contains 4,053,946 digits and was discovered November 14th. It belongs to a special class of rare prime numbers called Mersenne primes . The discovery marks only the 39th known Mersenne prime, named after Marin Mersenne , a 17th century French monk who studied the numbers. Mersenne primes are most relevant to number theory, but most participants join GIMPS simply for the fun of having a role in real research - and the chance of finding a new Mersenne prime. Cameron used a 800 MHz AMD T-Bird PC running part-time for 45 days to prove the number prime. He said, "A friend informed me that if I was going to leave my computer on all the time I should make use of that wasted CPU time. I put GIMPS on my PC because it does not interfere with my work on the computer. Finding the new prime was a wonderful surprise!"

13. Landon Curt Noll's Prime Pages
    Slowinski and Gage's discovery of the 34th mersenne prime Known mersenne primeDigits and Names; Caldwell's article on the discovery of 2 13466917 1;  
    http://www.isthe.com/chongo/tech/math/prime/

14. PROJECT GUTENBERG OFFICIAL HOME SITE -- Listing By AUTHOR
    David Slowinski.  
    http://promo.net/cgi-promo/pg/cat.cgi?&label=ID&ftpsite=ftp://ibiblio.or

15. Mersenne Prime - A Whatis Definition
    A Mersenne (also spelled Marsenne) prime is a specific type of prime number. It must be reducible to the form 2 to the nth power 1, where n is a prime number.  
    http://whatis.techtarget.com/definition/0%2C%2Csid9_gci788348%2C00.html
    
    Extractions: Search our IT-specific encyclopedia for: or jump to a topic: Choose a topic... CIO CRM Databases Domino Enterprise Linux IBM S/390 IBM AS/400 Networking SAP Security Solaris Storage Systems Management Visual Basic Web Services Windows 2000 Windows Manageability Advanced Search Browse alphabetically: A Mersenne (also spelled Marsenne) prime is a specific type of prime number . It must be reducible to the form 2 n - 1, where n is a prime number. The term comes from the surname of a French monk who first defined it. The first few known values of n that produce Mersenne primes are where n n n n n n n n n = 61, and n With the advent of computers to perform number-crunching tasks formerly done by humans, ever-larger Mersenne primes (and primes in general) have been found. The quest to find prime numbers is akin to other numerical searches done by computers. Examples are the decimal expansions of irrational number s such as pi (the circumference-to-diameter ratio of a circle) or e (the natural logarithm base). But the 'next' (unknown today, known tomorrow) prime is more difficult to find than the 'next' digit in the expansion of an irrational number. It takes the most powerful computer a long time to check a large number to determine if it is prime, and an even longer time to determine if it is a Mersenne prime. For this reason, Mersenne primes are of particular interest to developers of strong

16. Mersenne Primes: History, Theorems And Lists
    numbers. Definition When 2 n 1 is prime it is said to be a mersenne prime.It respectively)? And that in each case 2 n -1 was a mersenne prime? In  
    http://www.utm.edu/research/primes/mersenne/
    
    Extractions: History, Theorems and Lists Early History Perfect Numbers and a Few Theorems Table of Known Mersenne Primes The Lucas-Lehmer Test and Recent History ... Conjectures and Unsolved Problems See also Where is the next larger Mersenne prime? and Mersenne heuristics For remote pages on Mersennes see the Prime Links' Mersenne directory Primes: Home Largest Proving How Many? ... Mailing List Many early writers felt that the numbers of the form 2 n -1 were prime for all primes n , but in 1536 Hudalricus Regius showed that 2 -1 = 2047 was not prime (it is 23 89). By 1603 Pietro Cataldi had correctly verified that 2 -1 and 2 -1 were both prime, but then incorrectly stated 2 n -1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29. Sometime later Euler showed Cataldi's assertion about 31 was correct. Enter French monk Marin Mersenne (1588-1648). Mersenne stated in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2 n -1 were prime for n 31, 67, 127 and 257

17. The Prime Glossary: Mersenne Prime
    This pages contains the entry titled 'mersenne prime.' Come explore a new primeterm today! mersenne prime (another Prime Pages' Glossary entries).  
    http://primes.utm.edu/glossary/page.php?sort=Mersennes

18. Electric Word Prime Time
    George Woltman hopes his search for the largest known mersenne prime number will take on a life of its own. Wired News  
    http://www.wired.com/news/news/story/1654.html

19. The Prime Glossary: Great Internet Mersenne Prime Search
    prime numbers. This pages contains the entry titled 'Great Internetmersenne prime Search.' Come explore a new prime term today!  
    http://primes.utm.edu/glossary/page.php?sort=GIMPS

20. Mersenne Prime -- From MathWorld
    mersenne prime, A Mersenne number which is prime is called a Mersenneprime. Every mersenne prime gives rise to a perfect number.  
    http://mathworld.wolfram.com/MersennePrime.html
    
    Extractions: to be prime n must itself be prime . This is true since for composite n with factors r and s n = rs . Therefore, can be written as , which is a binomial number and can be factored. Every Mersenne prime gives rise to a perfect number . The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (Sloane's ) corresponding to n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (Sloane's If is a prime , then divides iff is prime . It is also true that prime divisors of must have the form where k is a positive integer and simultaneously of either the form or (Uspensky and Heaslet 1939). A prime factor p of a Mersenne number is a Wieferich prime iff . Therefore, Mersenne primes are not Wieferich primes . All known Mersenne numbers with p prime are squarefree . However, Guy (1994) believes that there are which are not squarefree Trial division is often used to establish the compositeness of a potential Mersenne prime. This test immediately shows

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