61. Perfect Numbers My Talk on Aliquot Parts mersenne primes The GREAT Internet mersenne prime SearchGIMPS Plot of mersenne primes Plot of mersenne prime Discoveries Perfect http://pw1.netcom.com/~hjsmith/Perfect.html

Perfect Numbers Select a Subtitle: What is a Perfect Number? My Talk on Aliquot Parts Mersenne Primes The GREAT Internet Mersenne Prime Search GIMPS ... Return to Harry's Home Page This page accessed times since June 7, 1997.

62. MersennePrime mersenne prime (English). Search for mersenne prime in NRICH PLUS maths.org Google. Definition level 2. A prime number which http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=1910

63. Rechenkraft.net :: Portal Translate this page Logo. Überblick. Name, Great Internet mersenne prime Search. Kategorie, Mathematik. Datum,Sprache, Meldung. 02.12.2003, 40th mersenne prime Announced. 02.12.2003, 40. http://217.160.138.71/project.php?id=gimps

64. Newseintrag Vom 02.12.2003 Translate this page Sprache, Titel, Externer Link40th mersenne prime Announced. Datum, 02.12.2003. ProjektGreat Internet mersenne prime Search, Liste, Ältere Meldung, Neuere Meldung. http://217.160.138.71/news.php?id=665

65. Four Years Of Mersenne Prime Searching KC Lemson (rss); Kristen Meston (rss); NBA Weblogs (rss); Ray Ozzie(rss). Four Years of mersenne prime Searching. Hmmm it s been http://blogs.msdn.com/tmeston/archive/2003/07/19/10286.aspx

load of Tosh Seymour Cray, do you know my name? posts - 117, comments - 231, trackbacks - 21 My Links Home Contact Login News My name is Tosh Meston. I am a Microsoft developer working on the Outlook Web Access team. Archives May, 2004 (4) April, 2004 (9) March, 2004 (11) February, 2004 (10) ... June, 2003 (15) Post Categories .NET (rss) Blogging (rss) ... (rss) Life Online Orkut Profile Photoblog / Moblog Sports InfiniteHoops.com (rss) InfiniteSoccer.com (rss) MSDN A Fond Farewell All Work and No Play Coming Up for Air Redmond's Gray Skies of Summer ... Standard Answers People I am reading .NET Weblogs (rss) Blogs.MSDN.com Dare Obasanjo ... Four Years of Mersenne Prime Searching Hmmm...  it's been four years of running GIMPS (Great Internet Mersenne Prime Search) software on my machines and in that time I've checked 117 exponents and the project has found 2 Mersenne prime numbers (prime numbers of the form 2 ^P I still think projects like these are a good use of spare cycles.  What distributed computing software do you run? posted on Saturday, July 19, 2003 12:36 AM Feedback No comments posted yet.

66. GIMPS Finds 40th Known Mersenne Prime KC Lemson (rss); Kristen Meston (rss); NBA Weblogs (rss); Ray Ozzie(rss). GIMPS finds 40th known mersenne prime. From www.mersenne.org http://blogs.msdn.com/tmeston/archive/2003/12/02/40814.aspx

load of Tosh Seymour Cray, do you know my name? posts - 117, comments - 231, trackbacks - 21 My Links Home Contact Login News My name is Tosh Meston. I am a Microsoft developer working on the Outlook Web Access team. Archives May, 2004 (4) April, 2004 (9) March, 2004 (11) February, 2004 (10) ... June, 2003 (15) Post Categories .NET (rss) Blogging (rss) ... (rss) Life Online Orkut Profile Photoblog / Moblog Sports InfiniteHoops.com (rss) InfiniteSoccer.com (rss) MSDN A Fond Farewell All Work and No Play Coming Up for Air Redmond's Gray Skies of Summer ... Standard Answers People I am reading .NET Weblogs (rss) Blogs.MSDN.com Dare Obasanjo ... GIMPS finds 40th known Mersenne Prime From www.mersenne.org On November 17, 2003 Michael Shafer's computer found the 40th known Mersenne prime, 2 -1! This number "weighs in" at a whopping 6,320,430 decimal digits! This is also the largest known prime number, surpassing GIMPS' last discovery by over 2 million digits. Here's the number in case you are interested. posted on Tuesday, December 02, 2003 4:31 PM Feedback No comments posted yet.

67. Mersenne Prime :: Online Encyclopedia :: Information Genius mersenne prime. Online Encyclopedia A mersenne prime is a prime numberthat is one less than a power of two. For example http://www.informationgenius.com/encyclopedia/m/me/mersenne_prime.html

Quantum Physics Pampered Chef Paintball Guns Cell Phone Reviews ... Science Articles Mersenne prime Online Encyclopedia A Mersenne prime is a prime number that is one less than a power of two Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist. It is currently unknown whether there is an infinite number of Mersenne primes. More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than an odd power of two; the notation M n n shows that M n can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; M n may be composite even though n is prime. For example, 2

68. Integer Lists: Mersenne Primes mersenne primeS. Mersenne The great mersenne prime race has been in progressnow for over 600 years and shows no sign of ending. Some http://users.argonet.co.uk/oundlesch/alists/mers.html

MERSENNE PRIMES Mersenne Primes are of the form , where p is itself prime. The search was on when it was noticed that most of the early primes worked, but 2^11-1 was not prime. The great Mersenne Prime race has been in progress now for over 600 years and shows no sign of ending. Some of the more reasonably sized numbers are given in this list together with the date of discovery Who was Mersenne? Return to Integer Lists menu : by pure brain-power - 2^11-1 = 2047 = 23 x 89 = 8191 .. .. .. .. .. .. .. [proved 1456] = 131071 .. .. .. .. .. .. .. [proved 1588, Cataldi] = 524287 .. .. .. .. .. .. .. [proved 1588, Cataldi] - 2^23-1 = 8388607 = 47 x 178481 - 2^29-1 = 536870911 = 233 x 1103 x 2089 = 2147483647 .. .. .. .. .. .. .. [proved 1772, Euler] - 2^37-1 = 137438953471 = 223 x 616318177 - 2^41-1 = 2199023255551 = 13367 x 164511353 8796093022207 = 431 x 9719 x 2099863 140737488355327 = 2351 x 4513 x 13264529 9007199254740991 = 6361 x 69431 x 20394401 -1 = 576460752303423487 = 179951 x 3203431780337 by mechanical calculators M 9: 2^ 61-1 [proved 1883, Pervushin] = 2.306 x 10^18 (4 s.f.) = 2305843009213693951

69. Mersenne Primes GIMPS Great Internet mersenne prime Search. Be a GIMPS Participant. Thoughouthistory, the largest known prime number is usually a mersenne prime. http://southern-alberta.com/pieis/play/math/mprime/Default.htm

PI EIS Playground Educational Internet Projects Mathematics GIMPS Great Internet Mersenne Prime Search Be a GIMPS Participant Join the rigorous search for Mersenne Primes. The search requires thousands of cpu hours to accomplish. Your mission, should you choose to accept it, is to select a range and assist with meeting the goal of testing every prime under 5,260,000 by the year 2000. Mersenne numbers are of the form 2^p - 1. Some of these numbers are prime. Thoughout history, the largest known prime number is usually a Mersenne prime. Download a program and database file Select a range and notify George F. Woltman (email: woltman@magicnet.net ) about your selection. Run the program and enter the range that you selected under the Range menu. Two months later, you're done! If required, you can stop the program. A Continue option under the Range menu can be invoked to resume your range test. Since the program runs at the lowest priority, it has little impact on response time. People with 486 machines can help with the search by factoring.

70. ARIA - GIMPS Project mersenne primes GIMPS Project. GIMPS, the Great Internet mersenne prime Search,was formed in January 1996 to discover new world-record-size mersenne primes. http://www.aria.uklinux.net/mersenne.php

Mersenne Primes - GIMPS Project Main Homepage Articles Printer Friendly Pages Software NURBS Toolbox Tips and Tricks Misc Mersenne Primes GIMPS , the Great Internet Mersenne Prime Search, was formed in January 1996 to discover new world-record-size Mersenne primes. GIMPS harnesses the power of thousands of small computers to search for primes of the form 2 P There are only 39 Mersenne primes known, and the GIMPS project has discovered five Mersenne primes so far. The 40th Mersenne Prime might have been found 17th Nov 2003, see GIMPS page. Confirmation will be early December. Date Name P 17th Nov 2003 14th Nov 2001 Michael Cameron 1st June 1999 Nayan Hajratwala 27th Jan 1998 Roland Clarkson 24th Aug 1997 Gordon Spence Nov 1996 Joel Armengaud The table below show the progress up the GIMPS ranks. Rank LL P90* Exponents Fact.P90 CPU yrs LL Tested CPU yrs* dmspink@aria.uklinux.net

71. Mersenne Primes Mersenne Numbers Primes. In the statements below p is an odd prime.All 2 p 1 have the form 8k+7 They also have the form pk+1, This http://home.earthlink.net/~usondermann/mer.html

In the statements below p is an odd prime. All p have the form They also have the form pk+1, This is from the Little Fermat Theorem where p is mod p therefore by adding 1 to both sides we get p is 1 mod p. This may be a clue to the factors of p that are composite. My guess is that pk+1 is a factor, k p p Here is a look at the Mersenne Composites Another obvious fact is that (p-1) is mod p, E.G. has a factor of Here is a another tidbit If p is prime then p is divisible by can be factored into p p since is also factorable by because the exponent is composite since p is prime and factors to p p or must divide p Here is a look at some Fermat Numbers Last updated: 5/26/98 Webmaster: Ulrich (Ulie) Sondermann usondermann@earthlink.net

72. Notable Properties Of Specific Numbers At MROB All such numbers are called Mersenne numbers, but most are not prime. If a Mersennenumber is prime (as is the case for 127) it is called a mersenne prime. http://home.earthlink.net/~mrob/pub/math/numbers-5.html

Notable Properties of Specific Numbers Back to page 4 Forward to page 6 Another cult number , notable (among other reasons) for being the same when turned upside down. Such numbers are called strobogrammatic . The last three years with this property were 1691, 1881 and 1961; the next isn't until 6009. The complete sequence of invertable numbers starts: 0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119, ... (Sloane's ). To make an invertible number, start with any combination of the digits 0, 1, 6, 8 and 9; then add the same set of digits in reverse order and inverted. If the middle two digits are 00, 11 or 88, you can make another invertible number by removing one of the duplicated digts. For example, starting with 68 we can make the two invertable numbers 6889 and 689. In 1961 there was actually an article in a mainstream math journal titled "Strobogrammatic Years". The Pell numbers are similar to the Fibonacci numbers and are generated by the formula A n = 2 A n-1 + A n-2 . The sequence runs: 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, ... (Sloane's

73. Perfect Numbers And Mersenne Primes Index Perfect numbers and mersenne primes. It Letting gives the conclusion.A prime number of the form is called a mersenne prime. As we http://web.usna.navy.mil/~wdj/book/node19.html

Next: Primality testing Up: Primes Previous: The Fundamental Theorem of Contents Index Perfect numbers and Mersenne primes It is remarkable that even at this ``elementary'' level there are many problems which are still unsolved. In this section, we mention one of the oldest unsolved problems in mathematics. Let be an integer and let For example, Definition 1.5.12 A perfect number is an integer such that , in other words, is the sum of its proper divisors. No odd perfect numbers are known. The following conjuecture may be the oldest unsolved problem in mathematics! Conjecture 1.5.13 Odd perfect numbers don't exist. It is known if an odd perfect number exists then it must be at least Lemma 1.5.14 An integer is an even perfect number if and only if , where is a prime. Though this result is often quoted as being due to Euler, it may have been known to Euclid. proof : We leave the ``if'' direction as an exercise. ``Only if'': Since is an even number, we can write , where is odd and . We know that , so let , with . Since , we have: which gives us: Therefore

74. Off The Kuff: New Mersenne Prime Found December 11, 2003. New mersenne prime found I love stories about big prime numbers. 17,when he glanced at the screen and saw New mersenne prime found. . http://www.offthekuff.com/mt/archives/002724.html

Off the Kuff Knowledge Is Good Contact me: kuff - at - offthekuff.com Main December 11, 2003 New Mersenne prime found I love stories about big prime numbers More than 200,000 computers spent years looking for the largest known prime number. It turned up on Michigan State University graduate student Michael Shafer's off-the-shelf PC. "It was just a matter of time," Shafer said. The number is 6,320,430 digits long and would need 1,400 to 1,500 pages to write out. It is more than 2 million digits larger than the previous largest known prime number. Shafer, 26, helped find the number as a volunteer on an eight-year-old project called the Great Internet Mersenne Prime Search. Tens of thousands of people volunteered the use of their PCs in a worldwide project that harnessed the power of 211,000 computers, in effect creating a supercomputer capable of performing 9 trillion calculations per second. Participants could run the mathematical analysis program on their computers in the background, as they worked on other tasks. Shafer ran an ordinary Dell computer in his office for 19 days until Nov. 17, when he glanced at the screen and saw "New Mersenne prime found."

75. Mersenne Primes next up previous contents Next Definition Up Related Topics PreviousDefinition Contents mersenne primes. Subsections Definition; http://www-maths.swan.ac.uk/pgrads/bb/project/node39.html

Next: Definition Up: Related Topics Previous: Definition Contents Mersenne Primes Subsections Definition if is prime ( ), then is prime B–ttcher 2002-08-07

76. 40th Mersenne Prime Is Discovered 40 th mersenne prime Is Discovered 40 th mersenne prime Yields LargestKnown Perfect Number Perfect Numbers and Large Numbers of Digits. http://members.aol.com/DrMWEcker/40th_mersenne_prime_is_discovered.htm

th Mersenne Prime Is Discovered: -1 Has 6,320,430 Digits; New Record-Holder For Largest Known Prime By Dr. Michael W. Ecker, REC Editor – December 2003 Michael Shafer, a 26-year-old graduate student in chemical engineering at Michigan State University, made mathematical history recently by discovering the largest known prime number to date. The new number is 2 – 1. It is a Mersenne prime, only the 40 th such number to have ever been found. He used his office computer as part of GIMPS , the Great Internet Mersenne Prime Search. The project has more than 60,000 volunteers from all over the world taking part. As reported here in REC , Project GIMPS has been linked to most or all of the discoveries of Mersenne primes in the past decade. Written out in the usual base-10 fashion, the new number is 6,320,430 digits long. It took just over two years to find using a distributed network of more than 200,000 computers. Michael found it on or around Nov. 17 th , but it took two weeks to get the independent verification. After confirmation, the result was widely announced in the first 2-3 days of December. It has been two years since the discovery of the previous record: the 39 th Mersenne prime, a number of a "mere" 4,053,946 digits.

77. [math-learn] 40th Mersenne Prime (Probably) Discovered By Jerry Becker mathlearn 40th mersenne prime (Probably) Discovered by Jerry Becker.Back to math-learn This would make it the 40th known mersenne prime. http://mathforum.org/epigone/math-learn/spyrdaldrau

[math-learn] 40th Mersenne Prime (Probably) Discovered by Jerry Becker Back to math-learn Subject: [math-learn] 40th Mersenne Prime (Probably) Discovered Author: jbecker@siu.edu Date: Fri, 28 Nov 2003 16:50:10 -0600 **************************** From http://mathworld.wolfram.com/news/2003-11-19/mersenne/ http://www.c1tracking.com/l.asp?cid=5511 http://us.click.yahoo.com/mOAaAA/3exGAA/qnsNAA/C2XolB/TM http://docs.yahoo.com/info/terms/ ... The Math Forum

78. 40th Mersenne Prime (Probably) Discovered By Deanna Marie De'Liberto 40th mersenne prime (Probably) Discovered by Deanna Marie De Liberto.Back number. This would make it the 40th known mersenne prime. http://mathforum.org/epigone/nysmath48/yanddwumhil

40th Mersenne Prime (Probably) Discovered by Deanna Marie De'Liberto Back to New York State "Grade 4" and "Grade 8" Math Assessments Subject: 40th Mersenne Prime (Probably) Discovered Author: ddeliberto@dsquared.to Date: Sun, 30 Nov 2003 15:30:18 -0500 This is a multi-part message in MIME format. =_NextPart_000_0060_01C3B756.DBEE3DC0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: 7bit **************************** From http://mathworld.wolfram.com/news/2003-11-19/mersenne/ http://www.dsquared.to Member of the Association of Test Publishers =================================================== Join "D2 Algebra Readiness Discussion Forum" now! http://www.topica.com/lists/D2Algebra/ http://mathworld.wolfram.com/news/2003-11-19/mersenne/ http://www.dsquared.to http://www.topica.com/lis= ... The Math Forum

79. The Great Internet Mersenne Prime Search - The Perl Journal, Winter 1997 The Great Internet mersenne prime Search. David Nicol. Until I computingdevices. Since 127 is prime, Lucas prime is also a mersenne prime. http://www.foo.be/docs/tpj/issues/vol2_4/tpj0204-0012.html

PREVIOUS TABLE OF CONTENTS NEXT The Great Internet Mersenne Prime Search David Nicol Until I became acquainted with GIMPS, the Great Internet Mersenne Prime Search (http://www.mersenne.org), the idle cycles on my workstation were spent entirely in wait states. No longer. Now my computer continuously runs a program that breaks new mathematical ground. Even as I write this, it's running in the intervals between my keystrokes, factoring large numbers as part of an Internet-wide coordinated effort to find the highest known prime number. In this article I'll describe how I use Perl not for factoring - a C program does that - but for automating communication with the computer (and person) in charge of GIMPS. Recently, the world has seen two large problems successfully attacked with massively distributed systems, an approach sometimes called "metacomputing." On June 18, the DES Challenge (DESCHALL), which teamed up computers around the world to decipher a message encrypted with a 56-bit RSA secret key, succeeded. On August 24, the GIMPS project demonstrated that 2 -1 is a prime number. Both DESCHALL and GIMPS are computer programs that ran with low priority on thousands of computers volunteered around the world.

80. What Is A Mersenne Prime? What is a mersenne prime? keywords=math, formula, algorithm Mersenne s When2^n1 is prime it is said to be a mersenne prime. Testing http://mediagods.com/glossary/What_is_a_Mersenne_prime.html

mediagods glossary Glossary How-To's ... Suggest What is a Mersenne prime? keywords=math, formula, algorithm Mersenne's Conjecture states that 2 raised to a power minus 1 is a prime. When 2^n-1 is prime it is said to be a Mersenne prime. Testing new primes is very computation intensive and is currently a popular application for distributed computing projects. News for Jun 6, 2004 The Macintosh News Network Briefly: dev tutorial, Ma... Ten ways Apple improved p... Apple store in Austin, TX... ... MediaGods Privacy Statement

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