PrimeGrid

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PrimeGrid
Logo of the PrimeGrid project, 2023.svg
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Original author(s) Rytis Slatkevičius
Initial releaseJune 12, 2005;19 years ago (2005-06-12) [1]
Development statusActive
Project goal(s)Finding prime numbers of various types
Software usedPRPNet, Genefer, LLR, PFGW
FundingCorporate sponsorship, crowdfunding [2] [3]
Platform BOINC
Average performance3,398.914 TFLOPS [4]
Active users2,330 (August 2022) [4]
Total users353,245 [4]
Active hosts11,504 (August 2022) [4]
Total hosts21,985 [4]
Website www.primegrid.com

PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads.

Contents

PrimeGrid awards badges to users in recognition of achieving certain defined levels of credit for work done. The badges have no intrinsic value but are valued by many as a sign of achievement. The issuing of badges should also benefit PrimeGrid by evening out the participation in the less popular sub projects. The easiest of the badges can often be obtained in less than a day by a single computer, whereas the most challenging badges will require far more time and computing power.

History

PrimeGrid started in June 2005 [1] under the name Message@home and tried to decipher text fragments hashed with MD5. Message@home was a test to port the BOINC scheduler to Perl to obtain greater portability. After a while the project attempted the RSA factoring challenge trying to factor RSA-640. After RSA-640 was factored by an outside team in November 2005, the project moved on to RSA-768. With the chance to succeed too small, it discarded the RSA challenges, was renamed to PrimeGrid, and started generating a list of the first prime numbers. At 210,000,000,000 [5] the primegen subproject was stopped.

In June 2006, dialog started with Riesel Sieve to bring their project to the BOINC community. PrimeGrid provided PerlBOINC support and Riesel Sieve was successful in implementing their sieve as well as a prime finding (LLR) application. With collaboration from Riesel Sieve, PrimeGrid was able to implement the LLR application in partnership with another prime finding project, Twin Prime Search (TPS). In November 2006, the TPS LLR application was officially released at PrimeGrid. Less than two months later, January 2007, the record twin was found by the original manual project. TPS has since been completed, and the search for Sophie Germain primes was suspended in 2024.

In the summer of 2007, the Cullen and Woodall prime searches were launched. In the Fall, more prime searches were added through partnerships with the Prime Sierpinski Problem and 3*2^n-1 Search projects. Additionally, two sieves were added: the Prime Sierpinski Problem combined sieve which includes supporting the Seventeen or Bust sieve and the combined Cullen/Woodall sieve. In the fall of the same year, PrimeGrid migrated its systems from PerlBOINC to standard BOINC software.

Since September 2008, PrimeGrid is also running a Proth prime sieving subproject. [6]

In January 2010 the subproject Seventeen or Bust (for solving the Sierpinski problem) was added. [7] The calculations for the Riesel problem followed in March 2010.

Projects

As of January 2023, PrimeGrid is working on or has worked on the following projects:

ProjectActive sieve project?Active LLR project?StartEndBest result
321 Prime Search (primes of the form 3 × 2n ± 1)NoYes30 June 2008Ongoing3 × 218196595 − 1, largest prime found in the 321 Prime Search project [8]
AP26 Search (Arithmetic progression of 26 primes)——27 December 200812 April 201043142746595714191 + 23681770 × 23# × n, n = 0, ..., 25 (AP26) [9]
AP27 Search (Arithmetic progression of 27 primes)——20 September 2016Ongoing224584605939537911 + 81292139 × 23# × n, n = 0, ..., 26 (AP27) [10]
Generalized Fermat Prime Search [11] [12]
(active: n = 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304 inactive: n = 8192, 16384)		Yes (manual sieving)—January 2012Ongoing19637361048576 + 1, largest known Generalized Fermat prime [13]
Cullen Prime SearchNoYesAugust 2007Ongoing6679881 × 26679881 + 1, largest known Cullen prime [14]
Message7No—12 June 2005August 2005PerlBOINC testing successful
Prime Sierpinski Problem NoYes10 July 2008Ongoing168451 × 219375200 + 1 [15]
Extended Sierpinski Problem NoYes7 June 2014Ongoing202705 × 221320516 + 1, largest prime found in the Extended Sierpinski Problem [16]
PrimeGenNo—March 2006February 2008—
Proth Prime SearchYesYes29 February 2008Ongoing7 × 25775996 + 1 [17]
Riesel Problem NoYesMarch 2010Ongoing9221 × 211392194 − 1, [18]
RSA-640 No—August 2005November 2005—
RSA-768 No—November 2005March 2006—
Seventeen or BustNoYes31 January 2010Ongoing10223 × 231172165 + 1
Sierpinski/Riesel Base 5 ProblemNoYes14 June 2013Ongoing213988×54138363 − 1, largest prime found in the Sierpinski/Riesel Base 5 Problem
Sophie Germain Prime SearchNoNo16 August 2009February 20242618163402417 × 21290000 − 1 (2p − 1 = 2618163402417 × 21290001 − 1), the world record Sophie Germain prime; [19] and 2996863034895 × 21290000 ± 1, the world record twin primes [20]
Twin prime SearchNo—26 November 200625 July 200965516468355 × 2333333 ± 1 [21]
Woodall Prime SearchNoYesJuly 2007Ongoing17016602 × 217016602 − 1, largest known Woodall prime [22]
Generalized Cullen/Woodall Prime SearchYesYes22 October 2016Ongoing2525532 × 732525532 + 1, largest known generalized Cullen prime [23]
Wieferich Prime Search——2020 [24] 2022—
Wall-Sun-Sun Prime Search——20202022—

321 Prime Search is a continuation of Paul Underwood's 321 Search which looked for primes of the form 3 · 2n − 1. PrimeGrid added the +1 form and continues the search up to n = 25M.

Primes known for 3 · 2n + 1 occur at the following n:

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 2478785, 5082306, 7033641, 10829346, 16408818 (sequence A002253 in the OEIS )

Primes known for 3 · 2n − 1 occur at the following n:

0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414, 6090515, 11484018, 11731850, 11895718, 16819291, 17748034, 18196595 (sequence A002235 in the OEIS )

PRPNet projects

ProjectActive?StartEndBest result
27 Prime SearchNo—March 2022 [25] 27 × 27046834 + 1, largest known Sierpinski prime for b = 2 and k = 27
27×28342438 − 1, largest known Riesel prime for b = 2 and k = 27 [26]
121 Prime SearchNo—April 2021 [27] 121 × 29584444 − 1, largest known Sierpinski prime for b = 2 and k = 121
121 × 24553899 − 1, largest known Riesel prime for b = 2 and k = 121 [28]
Extended Sierpinski problem No—201490527 × 29162167 + 1 [29]
Factorial Prime SearchYes—Ongoing147855! − 1, 5th largest known factorial prime
Dual Sierpinski problem (Five or Bust)No—All were done (all PRPs were found)29092392 + 40291
Generalized Cullen/Woodall Prime SearchNo—2017 [30] 427194 × 113427194 + 1, then largest known GCW prime [31]
Mega Prime SearchNo—201487 × 23496188 + 1, largest known prime for k = 87
Primorial Prime SearchYes2008 [32] Ongoing3267113# − 1, largest known primorial prime [33]
Proth Prime SearchNo20082012 [34] 10223 × 231172165 + 1, largest known Proth prime
Sierpinski Riesel Base 5No2009 [35] 2013 [36] 180062 × 52249192 − 1
Wieferich Prime SearchNo2012 [37] 2017 [38] 82687771042557349, closest near-miss above 3 × 1015
Wall-Sun-Sun Prime SearchNo2012 [37] 2017 [38] 6336823451747417, closest near-miss above 9.7 × 1014

Accomplishments

AP26

One of PrimeGrid projects was AP26 Search which searched for a record 26 primes in arithmetic progression. The search was successful in April 2010 with the finding of the first known AP26:

43142746595714191 + 23681770 · 23# · n is prime for n = 0, ..., 25. [39]
23# = 2·3·5·7·11·13·17·19·23 = 223092870, or 23 primorial, is the product of all primes up to 23.

AP27

Next target of the project was AP27 Search which searched for a record 27 primes in arithmetic progression. The search was successful in September 2019 with the finding of the first known AP27:

224584605939537911 + 81292139 · 23# · n is prime for n = 0, ..., 26. [40]
23# = 2·3·5·7·11·13·17·19·23 = 223092870, or 23 primorial, is the product of all primes up to 23.

PrimeGrid is also running a search for Cullen prime numbers, yielding the two largest known Cullen primes. The first one being the 14th largest known prime at the time of discovery, and the second one was PrimeGrid's largest prime found 6679881 · 26679881 + 1 at over 2 million digits. [41]

On 24 September 2022, PrimeGrid discovered the largest known Generalized Fermat prime to date, 19637361048576 + 1. This prime is 6,598,776 digits long and is only the second Generalized Fermat prime found for n = 20. It ranks as the 13th largest known prime overall. [42]

Riesel Problem

As of 13 December 2022, PrimeGrid has eliminated 18 values of k from the Riesel problem [43] and is continuing the search to eliminate the 43 remaining numbers. 3 values of k are found by independent searchers.

Primegrid worked with the Twin Prime Search to search for a record-sized twin prime at approximately 58,700 digits. The new world's largest known twin prime 2003663613 × 2195000 ± 1 was eventually discovered on January 15, 2007 (sieved by Twin Prime Search and tested by PrimeGrid). The search continued for another record twin prime at just above 100,000 digits. It was completed in August 2009 when Primegrid found 65516468355 × 2333333 ± 1. Continued testing for twin primes in conjunction with the search for a Sophie Germain prime yielded a new record twin prime in September 2016 upon finding the number 2996863034895 × 21290000 ± 1 composed of 388,342 digits.

As of 22 April 2018, the project has discovered the four largest Woodall primes known to date. [44] The largest of these is 17016602 × 217016602 − 1 and was found in 21 March 2018.[ citation needed ] The search continues for an even bigger Woodall prime. PrimeGrid also found the largest known generalized Woodall prime, [45] 563528 × 13563528 − 1.

Media coverage

PrimeGrid's author Rytis Slatkevičius has been featured as a young entrepreneur in The Economist . [46]

PrimeGrid has also been featured in an article by Francois Grey in the CERN Courier and a talk about citizen cyberscience in TEDx Warwick conference. [47] [48]

In the first Citizen Cyberscience Summit, Rytis Slatkevičius gave a talk as a founder of PrimeGrid, named Finding primes: from digits to digital technology, [49] relating mathematics and volunteering and featuring the history of the project. [50]

Related Research Articles

<span class="mw-page-title-main">Great Internet Mersenne Prime Search</span> Volunteer project using software to search for Mersenne prime numbers

The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.

<span class="mw-page-title-main">Prime number</span> Number divisible only by 1 or itself

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

Grid computing is the use of widely distributed computer resources to reach a common goal. A computing grid can be thought of as a distributed system with non-interactive workloads that involve many files. Grid computing is distinguished from conventional high-performance computing systems such as cluster computing in that grid computers have each node set to perform a different task/application. Grid computers also tend to be more heterogeneous and geographically dispersed than cluster computers. Although a single grid can be dedicated to a particular application, commonly a grid is used for a variety of purposes. Grids are often constructed with general-purpose grid middleware software libraries. Grid sizes can be quite large.

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In number theory, a Woodall number (Wn) is any natural number of the form

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References

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  7. ↑ John. "PrimeGrid forum: Seventeen or Bust and the Sierpinski Problem". PrimeGrid. Archived from the original on 2011-09-26. Retrieved 2011-09-19.
  8. ↑ "PrimePage Primes: 3·2^18196595 - 1". t5k.org. Archived from the original on 23 January 2022. Retrieved 12 March 2023.
  9. ↑ "PrimeGrid's AP26 Search" (PDF). PrimeGrid. Archived (PDF) from the original on 2011-09-26. Retrieved 2011-09-19.
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  22. ↑ "PrimeGrid's Woodall Prime Search" (PDF). PrimeGrid. Archived (PDF) from the original on 2021-01-21. Retrieved 2019-07-28.
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  37. 1 2 "Welcome to a week of Wieferich and Wall-Sun-Sun". PrimeGrid. Archived from the original on 2013-08-17. Retrieved 2013-07-03.
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  49. ↑ Rytis Slatkevičius (2010-09-02), Finding primes: from digits to digital technology, archived from the original on 2010-09-15, retrieved 2010-12-03
  50. ↑ Rytis Slatkevičius (2010-08-13), Giant Prime Numbers, archived from the original on 2011-07-08, retrieved 2010-12-03
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