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Mathematics  
The current largest known Mersenne prime number is
2^{20,996,011}  1 (found on November 17, 2003, and containing
6.3 million digits). This is the largest known prime number. Help find the
next one in the Great Internet Mersenne Prime
Search (GIMPS). Unix users can participate in GIMPS using precompiled clients or source code at Ernst Mayer's site and the manual testing forms at the PrimeNet server. Version 23.5 of the client is available as of July 4, 2003. Pentium 4 owners will see speed improvements of 25% with this version, and Athlon, Duron, Pentium 3, and Celeron 2 owners will see speed improvements of 38%. Beta version 23.7 of the client is available as of September 6, 2003. It fixes a torture test bug. See a complete list of features in this version. Join a discussion group about this project. 
ongoing 


Search for different kinds of prime numbers at Yves Gallot's Proth Search Page. Submit new primes to the Top 5000 Primes list. Version 7.1 of Proth is available as of October 29, 2002. See the list of prime numbers that I've found. 
ongoing 


Help find new factors of Cunningham numbers at ECMNET.  ongoing 


Search for the next prime of the form n!+1 (or of the form n!1) using primeform. You can reserve and submit ranges for either project through the website.  ongoing 


Find Minimal Equal
Sums of Like Powers using
Euler2000,
available on the download
page. The client automatically downloads ranges of numbers to work on.
On February 6, 2003, a project member discovered the largest (6,2,5) result above 60,000. On December 8, 2002, a project member found a new upper limit for Taxicab(6): Taxicab(6) <= 24153319581254312065344, since 24153319581254312065344 = 28906206^{3} + 582162^{3} = 28894803^{3} + 3064173^{3} = 28657487^{3} + 8519281^{3} = 27093208^{3} + 16218068^{3} = 26590452^{3} + 17492496^{3} = 26224366^{3} + 18289922^{3}. The Taxicab problem isn't a part of the Minimal Equal Sums of Like Powers project, but this is a big discovery nonetheless. Version 4.21b of the client is available as of September 30, 2002. This version handles reserved work ranges better, points to the new eulernet.org domain automatically, and on the server side, allows for simultaneous client connections. Note: you should upgrade from version 4.18 and earlier to fix a significant client bug in those versions. Note: 4.21b has a bug which prevents the client from connecting to the server if you try to reserve more than the maximum 100 ranges. Use temporary version 4.21c to fix the bug. Note: the project server's old ISP and domain (euler.myip.org) are unreachable as of October 15, 2002. Please use the domain eulernet.org. 
ongoing: Resta 3 (6,1,6): 100.0% (completed October 23, 2002) Resta 6 (6,2,5): 65.49% 


Search for factors of 2^(2^611)1, a double Mersenne number, in the MM61 project. Download and test the client, then email the project coordinator to reserve a range of numbers to test.  ongoing: 7,840 ranges done, 2,160 to do 


Find 3x+1 class records in the
3x+1 Problem
project. This project attempts to find ever higher 3x+1 class records.
The client, which will work on any PC/Windows platform, and the instructions
for joining the project, are
here.
Note: the client takes about 6 weeks to finish one block on a 400MHz
CPU.
The project completed its first goal, processing 20,000 blocks, on November 19, 2002. The project found a new glide record in December, 2002. This is the first glide record found in almost a year. The record occurs at 180352,746940,718527, and the new glide is 1575, an improvement of 104 over the previous record. The project found a new Path record in March, 2003. The previous Path record was found in late 2002. The record occurs at 212581,558780,141311, (or +/ 189.2^{50}) and it reaches a maximum of 4353,436332,008631,522202,821543,171376. See the project's progress. 
ongoing: all blocks up to 29,840 complete; all class records below 1,900 known 


The
pi(x) project calculates pi(x), for very large values of x. It most
recently calculated pi(x) for x=4*10^{22}. You can
contribute to the calculations for pi(x) for x=10^{23}.
Version 1.5 of the client is available as of February 17, 2001. The project is on hold as of April 15, 2002, while a problem in the software is resolved. 
on hold 


Help the Distributed Search for Fermat Number Divisors project find unique Fermat Number factors.
This site is also available in Italian , Russian , and German . Version 4.1 of the client is available as of September 7, 2001. The latest factor, 1,054,057^{.}2^{8,300} + 1 divides F_{8,298} was discovered by Craig Kitchen on November 1, 2003. 250 total Fermat factors are known: 8 have been found in 2003. Join the project discussion group or an independent discussion group about Fermat numbers. 
ongoing 


The
PCP@Home project looks for short cases of
Post's Correspondence
Problem with large
shortest solutions. This theoretical computer science problem has been in
existence since 1946. It demonstrates undecidability: "a problem that cannot
be solved for all cases by any algorithm whatsoever." Finding PCPs in this
project will help define "decidability criteria for bounded PCP classes."
To participate in the project, download a precompiled, statically linked executable for Linux ELF, FreeBSD ELF, Solaris 5.6, or Windows (you can also download and compile the source code), and also download a perl script called PcpSieve.pl which runs the executable, scans the output for record solutions, and emails the solutions to the project coordinator (you can also run the executable manually, search the output manually and email any record solutions you find). Note to Windows users: the Windows client was compiled by Michael Keppler of Rechenkraft.net. He says that it has a serious memory leak, and that you may need to kill it and restart it every day. If anyone knows how to debug Windows application memory leaks, please contact him. This site is also available in German. 
ongoing 


Find generalized Fermat prime numbers in the
Generalized Fermat Prime Search. This project uses the
Proth program or the
GFNSieve21 program to find these numbers. Unixusers and other users
can compile the
C source code of "GeneFer" for the project and you can directly check
a presieved range with
it. Version 1.2 of this code is available as of June 10, 2002.
On January 6, 2003, Daniel Heuer discovered the largest known Generalized Fermat prime 1483076^{65536}+1 (404,434 digits), with GFNSieve+Proth, beating his previous record from October 8, 2002. "This number is the new largest known prime which is not a Mersenne prime, and the 6th largest known prime." On February 16, 2003, Michael Angel discovered the first prime of the form b^{217} + 1: 62722^{131072} + 1 (628,808 digits). This number is the 5th largest known prime. On February 21, 2003, the project completed the whole range 22,200,000 for exponent 32768. It found 35 primes in this range. On March 26, 2003, Franz Hagel discovered the 20th Generalized Fermat prime of the form b^{65536} + 1: 357868^{65536} + 1 (363,969 digits). On July 12, 2003, Michael Angel discovered the second known prime of the form b^{217} + 1: 130816^{131072} + 1. On August 22, 2003, Daniel Heuer discovered the largest known Generalized Fermat prime: 1176694^{217} + 1. This 795,695 digit number is now the 5th largest known prime. On September 22, 2003, Daniel Heuer discovered the new largest known Generalized Fermat prime: 1372930^{217} + 1. This 804,474 digit number is now the 5th largest known prime. Join a discussion group about prime numbers. 
ongoing 


Find generalized Woodall numbers in the Generalized Woodall Numbers project. This project uses the Proth program to find these numbers.  ongoing 


Help search for the world's largest Proth prime number
in PSearch. A
Proth
prime is a prime number of the form Participants in the project should have at least a 600 Mhz PC. To join the project, first download George Woltman's PRP software for Windows or Linux. Then send email to William Garnett with your CPU type and speed and your operating system, and he will send you instructions for participating. Join a discussion group about the project. 
ongoing 


Help find the smallest
Sierpinski number in
Seventeen or Bust, a distributed attack on
the Sierpinski problem. The project looks for
Proth
prime numbers in which, for a number k, if every possible choice
of n results in a composite (nonprime) Proth number N,
k is a Sierpinski number.
The project began its k=33661 project on November 21, 2002, and fifteen additional projects on November 23, 2002. It has found the following primes:
To participate in the project, sign up for an account, download the client, add your account name to the client configuration, and run it. The client does Proth tests on individual numbers. Each number should take a few hours to test on an average machine. When the project server assigns you a number, it waits for up to 10 days for you to return your search results, and reassigns the number to someone else if it doesn't receive your results within that time. The client supports users behind firewalls and proxy servers. Version 1.11 of the client is available for Windows as of November 2, 2003. This version fixes a significant bug for Pentium 4 users. Version 1.0.2 is available for Linux, BeOS and FreeBSD as of November 27, 2002. Note: as of October 5, 2002, results from version 0.9.0 or earlier of the client are not accepted by the server. Please upgrade if you are running an older version. Windows users with dualCPU machines can download a special copy of the 0.9.9 Windows client which will allow them to run the client on both CPUs. Seventeen or Bust also has a supporting project to sieve numbers for the main project: sieving finds n numbers with small factors and removes them from the pool of prime number candidates which need to be tested by Seventeen or Bust. Two clients are available for sieving: SoBSieve (for Windows) and NBeGone (for multiple platforms). To reserve a range of numbers to sieve, post a message to the sieve coordination thread. Then submit the results from the range to the "sieve numbers" page mentioned above. 
11 primes remaining to be found; 225,235 Proth tests completed 


Find factorizations of cyclotomic numbers at
Factorizations
of Cyclotomic Numbers. This site doesn't appear to be
organized as an official distributed computing project and doesn't have
any precompiled client software or explicit instructions for participating,
so it is probably best suited for people who understand the Mathematical
principles behind the project and how to compile source code.
The Phi(92) series was completely factored by November 2, 2002: the last composite number was factored by Tetsuya Kobayashi on that date. Katsuyuki Okeya finished the Phi(61) and Phi(122) series on December 30, 2002. The Phi(69) series was been completely factored by January 12, 2003: the last composite number was factored by Alexander Kruppa. To participate in the project, you can download and compile a GMP or UBASIC factorization program, view a page of reserved numbers, then select a range of numbers to factorize and send email to Hisanori Mishima with the range information. Read a paper about cyclotomic polynomials and prime numbers by Yves Gallot. 
ongoing 


Help find prime numbers for
the dual Sierpinski problem
search. The project is trying to find a prime in each sequence of
integers of the form k+2^{n} (fixed k) for which no prime has yet
been found. The project is coordinated by Payam Samidoost, an active
researcher of Fermat numbers. Contact Payam to reserve numbers to check
and to submit your results.
The project uses George Woltman's PRP software (available for Windows and Linux). Instructions for downloading and using the software are listed here. 
ongoing 


Help verify Riemann's hypothesis in
ZetaGrid. This
hypothesis was formulated in 1859 and states that "all nontrivial zeros of
the Riemann zeta function (see the website) are on the critical line
(1/2+it where t is a real number)." No one has been able
to prove the hypothesis in 140 years. It is now considered one of
the most important problems of modern mathematics. The project offers
financial prizes.
See close
zeros found by the project. See the
current results of
the project.
The client runs as a Windows screensaver or service or as a commandline application. The commandline version displays a configurable amount of information about what it is doing. You must have Java Runtime Environment 1.2.2 or higher installed to use the client. It only needs to be connected to the Internet to receive work or send results. Version 1.8.6 of the client is available as of December 1, 2003: it adds some new features, several bug fixes, and a 10% improvement in performance. A Control Center client is also available for Windows: it allows you to monitor multiple CPUs and estimates the time remaining to complete an active work unit. A beta version of the ZetaGrid library is available for X86 Linux as of May 3, 2003. It is about 30% faster than the previous version. Note to users of the commandline client. You must download both the zeta_base.zip and zeta_platform.zip files to use this version of the client. Edit zeta.cfg to specify your user information and how you want the client to use your system. Next, download the zeta.cmd or zeta.sh startup script and edit it to define your JAVA_HOME and proxy server (if you have one) variables. See research papers about this project and its results. 
ongoing: 546 billion results returned 


Join the
Goldbach Conjecture
Verification to
help verify the conjecture through 1e18 (it is currently verified
through 1e16). The Goldbach conjecture is "one of the oldest unsolved problems
in number theory. ...it states that every even number larger than two can be
expressed as the sum of two prime numbers."
The client software consists of a server application which must be run on a GNU/Linux system with a version 2.4 or later kernel, and a client application which may be run on the same GNU/Linux system or on other GNU/Linux or Windows NT/2000/XP systems which can communicate with the server application over an intranet. To participate in the project, send email to Tomás Oliveira e Silva, the project coordinator, with information about the machine(s) on which you will run the server and client applications, and he will send you more information about how to participate. 
ongoing 


Help the
pi(x) Table Project construct a very large table of values of pi(x) for
large values of x. The table will allow people to study the behavior of
the pi(x) function in large ranges, a study which has never before been
possible. The project might also find the first known change of sign of the
function pi(x)Li(x). The first phase of the project computed pi(x) for
1.e16 < x < 1.e17. The current phase is computing pi(x) for 1.e17 <
x < 1.e18.
To participate in the project, download the DOS executable fastpix11.exe from the project website, reserve ranges of numbers through the website, process the ranges, and submit your results to the website. You may reserve a range for no longer than two weeks. 
unknown 


Help
The Riesel Problem project prove that k=509203 is the smallest
Riesel
Number. See the project's results on the
search status page.
To participate, download the proth.exe client, view reserved ranges on the checked out and progress page, then reserve a range (and submit your results) on the range reservation page. 
ongoing 


Help find primes of the form 3 * 2^{n}  1
in the 3*2^n1 Search.
This project builds on the work of the project
to find primes of the form k * 2^{n}  1 for k < 300. For
3 * 2^{n}  1, n is known up to 164,987. This project
will initially look for new ns between 191,600 and 1,000,000.
On April 11, 2003, the project found prime 3 * 2^{234760}1 (70,671 digits). To participate, download one of the following software clients: LLR, PFGW, PRP (see download links in the discussion forum) and send email to Paul Underwood. He will send you instructions on how to participate and will give you blocks to process. Note: this project only has a discussion forum and no website. 
18% 


NFSNET uses "the
Number Field Sieve to find the factors of increasingly large numbers."
You can participate in the project by following the instructions on the
join page. See the project's
latest news.
On November 10, 2002, NFSNET completed the factorization of W(668), a 204digit special number field sieve (SNFS). The client is commandline, with a GUI wrapper client available. It is better for users with permanent Internet connections. It supports users behind firewalls but not users behind proxy servers. Release Candidate 1 of the client is available as of May 12, 2003. Results so far:
Subscribe to the project mailing list. 
ongoing 


Join the
Search
for Multifactorial Primes. This project continues work started by
Ray Ballinger to find
multifactorial prime numbers, primes of the form n!!+/1,
n!!!+/1, n!!!!+/1, searching for all primes up to 10,000
digits for each multifactorial type.
On June 7, 2003, Jiong Sun finished !18+/1 to 150000. Participation instructions are at the top of the project page. Basically, you email the project coordinator to reserve a type, then use the multisieve and pfgw Windows applications to sieve the range and find primes in it, then submit your results to the project coordinator. 
ongoing 


Join the
GMPECM factoring
Lucas numbers project. This project is "helping to complete the factorization
tables with
Fibonacci and
Lucas numbers which are maintained by Blair Kelly
... focusing only on composites of Lucas numbers (composites are the remains
left over when a number is successfully divided by a prime number and the
remaining is not prime)." The project hopes to find all prime numbers that
make up each Lucas number. It tries "to find prime factors up to 55 digits,
this ... upper limit of the GMPECM program."
See results of the project. Lucas numbers currently being worked on:
The Windows client includes different cores optimized for various platforms such as AMD Athlon and Intel Pentium 4. The client software source code is also available, so you can compile it for Linux or your favorite flavor of Unix. A server application is also available, to make it easier for you to run the client on multiple systems in an Intranet. The latest versions of the Client and Server applications are available as of December 7, 2003. 
ongoing 


Help find primes of the form 15k *
2^{n}  1 in the
15k*2^n1 club. The project is looking for 15k which produce many primes
n: "in general when k has many small factors, n is more likely to
produce a prime because it cant have those same factors."
To participate, download the following software clients: NewPGen sieve, LLR prime tester (see download links at the project site), choose a 15k value and send email to TTcreation@aol.com along with the name or screen name you would like to use for the project: detailed participation instructions are in the discussion forum. Join a discussion forum about the project. 
ongoing 


Help verify the
Collatz
Conjecture for larger values in the
Collatz Conjecture project, hosted
on the Grid on Tap computing platform.
This project is helping to test the Grid on Tap platform.
Collatz Phase 1 tested n from 1 to 99,999,999,999. It began on August 22, 2003, and ended on September 3, 2003. Collatz Phase 2 tested n from 100,000,000,000350,010,009,999. It began on September 2, 2003 and ended on September 10, 2003. Collatz Phase 3 tested n from 350,010,009,999850,060,009,999. It began on September 10, 2003. Collatz Phase 4 tested n from 850,060,009,9991,850,160,009,999. Collatz Phase 5 tested n from 1,850,160,009,99911,851,160,009,999. Collatz Phase 6 tested n from 11,851,160,009,99921,852,160,009,999. Collatz Phase 7 tested n from 21,852,160,009,99971,857,160,010,000. Collatz Phase 8 is testing n from 71,857,160,010,000121,862,160,010,001. Information about downloading and using the Grid on Tap client is available at the Grid on Tap platform site. 
Collatz Phase 8: 1.73% 


Find factors of the Mersenne number
M(3326400) = 2^{3326400}  1 in
ElevenSmooth.
To participate in the project, download the ECMclient application and configure it according to the directions on the download page (if you already have the ECM or ECMclient application installed, you only need to reconfigure it to use server=wblipp.dynu.com and port=8194). Unix users can follow instructions to create an ECM client for Unix. Once ECMclient is configured, it contacts the ElevenSmooth project server to get work units and to return results. It processes a work unit for 30 minutes by default, but you can change the processing time by changing the maxfreq parameter. The project supports users behind firewalls and possibly proxy servers. It supports modem users with a little bit of work. See the help page for information about using firewalls, proxy servers, and modems. The project also has a Special Project subproject for users who have contributed at least one full week to the main ECM project. The Special Project "uses GIMPS' program Prime95 to work on all primitives of M(3326400) simultaneously. If any ECM work is going to be done on the largest composites, Prime95 is much faster. The subfactor composites are then tested 'for free.' However, even with Prime95, it takes a long time to run ECM curves on large numbers." Users who qualify for this project will be invited by email to join it. Join a discussion forum about the project. 
ongoing 


Help the
Riesel Sieve's efforts to solve the
Riesel problem by
"removing prime candidates for the remaining 99 K's from a huge .dat file
containing over 11 million k/n pairs. The .dat file contains n values from a
current low of about 400,000 to a high of 20,000,000 per remaining K.
Individual sieving efforts per single K can take two weeks to a month to
sieve to a sufficient level. This coordinated sieving effort will allow us to
sieve 100 times deeper and much quicker. No more sieving to 3T and stopping
in frustration as the hours per factor mount, now we can go to 300T and
beyond..." Before the project began, there were 101 candidate K's.
See the status of all
remaining K values.
On October 15, 2003, the project found a prime number: 212893 * 2^{730387}  1. On October 22, 2003, it found another prime number: 458743 * 2^{547791}  1. On December 22, 2003, it found another prime number: 261221 * 2^{689422}  1. To participate in the project, view instructions in the download discussion forum. Basically you download the rieselsieve.exe executable and the riesel.dat data file (about 23 MB compressed and 89 MB fully expanded). Then you reserve a range of numbers, process them with rieselsieve.exe, and submit your resulting factors. The riesel.dat file is updated regularly and gets smaller as prime candidates are removed. The latest version of the client is available for Windows and Linux as of December 13, 2003. This version is about 80% faster than previous versions. A client should be available for FreeBSD soon. The riesel.dat file was last updated on December 22, 2003. Join a discussion forum about the project. 
3,919,139 k/n pairs eliminated; 7,813,332 k/n pairs remaining 


Help The Prime Sierpinski Problem find the smallest prime Sierpinski number. "The smallest known prime Sierpinski number is k=271129. Finding a prime of type k * 2^{n} + 1 for all prime k < 271129 will be sufficient to prove that 271129 is the smallest prime Sierpinski number." The project found its first prime number, 87743 * 2^{212565} + 1, in December, 2003. It found its second prime number, 224027 * 2^{273967} + 1, on December 12, 2003. To participate in the project, choose a k from the status page, then email the project coordinator to reserve the k. Then use the newpgen and PRP applications (see links on the project website: clients are available for Windows and Linux) to find primes. If you find a prime, email the project coordinator so he can delete the k on the status page. 
ongoing 


Help P.I.E.S (Prime Internet Eisenstein Search) find large prime Generalised Eisenstein Fermat numbers numbers. The project's main goal is to study the properties of these numbers, and to that it needs to find some prime numbers. See the project's results. To participate in the project, send email to the project coordinator, user "thefatphil" at host "yahoo.co.uk", to let him know you're interested. Then download the client and follow the instructions on the download page for running it. Version 0.7 of the client is available for Windows. Version 0.8 is available for Linux, FreeBSD, AIX and Irix. 
ongoing 

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