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#886113

32-526: Let π ( N ) {\displaystyle \pi (N)} denote the number of primes less than or equal to N {\displaystyle N} . If A {\displaystyle A} is a subset of the prime numbers such that then for all positive integers k {\displaystyle k} , the set A {\displaystyle A} contains infinitely many arithmetic progressions of length k {\displaystyle k} . In particular,

64-496: 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. PrimeGrid started in June 2005 under the name Message@home and tried to decipher text fragments hashed with MD5 . Message@home

96-514: Is less than y (an irreflexive relation ). Similarly, using the convention that ⊂ {\displaystyle \subset } is proper subset, if A ⊆ B , {\displaystyle A\subseteq B,} then A may or may not equal B , but if A ⊂ B , {\displaystyle A\subset B,} then A definitely does not equal B . Another example in an Euler diagram : The set of all subsets of S {\displaystyle S}

128-486: Is vacuously a subset of any set X . Some authors use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols ⊆ {\displaystyle \subseteq } and ⊇ . {\displaystyle \supseteq .} For example, for these authors, it

160-734: Is a proper subset of B . The relationship of one set being a subset of another is called inclusion (or sometimes containment ). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B . A k -subset is a subset with k elements. When quantified, A ⊆ B {\displaystyle A\subseteq B} is represented as ∀ x ( x ∈ A ⇒ x ∈ B ) . {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).} One can prove

192-544: 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

224-803: Is called its power set , and is denoted by P ( S ) {\displaystyle {\mathcal {P}}(S)} . The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P ( S ) {\displaystyle {\mathcal {P}}(S)} defined by A ≤ B ⟺ A ⊆ B {\displaystyle A\leq B\iff A\subseteq B} . We may also partially order P ( S ) {\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B  if and only if  B ⊆ A . {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.} For

256-533: Is equivalent to A ⊆ B , {\displaystyle A\subseteq B,} as stated above. If A and B are sets and every element of A is also an element of B , then: If A is a subset of B , but A is not equal to B (i.e. there exists at least one element of B which is not an element of A ), then: The empty set , written { } {\displaystyle \{\}} or ∅ , {\displaystyle \varnothing ,} has no elements, and therefore

288-563: Is the constant The result was made unconditional by Green–Tao and Green–Tao–Ziegler. Green and Tao's proof has three main components: Numerous simplifications to the argument in the original paper have been found. Conlon, Fox & Zhao (2014) provide a modern exposition of the proof. The proof of the Green–Tao theorem does not show how to find the arithmetic progressions of primes; it merely proves they exist . There has been separate computational work to find large arithmetic progressions in

320-489: Is the product of the prime numbers up to 23, more compactly written 23# in primorial notation. On May 17, 2008, Wróblewski and Raanan Chermoni found the first known case of 25 primes: On April 12, 2010, Benoît Perichon with software by Wróblewski and Geoff Reynolds in a distributed PrimeGrid project found the first known case of 26 primes (sequence A204189 in the OEIS ): In September 2019 Rob Gahan and PrimeGrid found

352-405: Is true of every set A that A ⊂ A . {\displaystyle A\subset A.} (a reflexive relation ). Other authors prefer to use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate proper (also called strict) subset and proper superset respectively; that is, with

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384-656: 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. In January 2010

416-666: 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 × 2 ± 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 × 2 ± 1 . Continued testing for twin primes in conjunction with

448-500: The k -tuple from { 0 , 1 } k , {\displaystyle \{0,1\}^{k},} of which the i th coordinate is 1 if and only if s i {\displaystyle s_{i}} is a member of T . The set of all k {\displaystyle k} -subsets of A {\displaystyle A} is denoted by ( A k ) {\displaystyle {\tbinom {A}{k}}} , in analogue with

480-706: 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. 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

512-567: 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

544-563: The entire set of prime numbers contains arbitrarily long arithmetic progressions. In their later work on the generalized Hardy–Littlewood conjecture , Green and Tao stated and conditionally proved the asymptotic formula for the number of k tuples of primes p 1 < p 2 < ⋯ < p k ≤ N {\displaystyle p_{1}<p_{2}<\dotsb <p_{k}\leq N} in arithmetic progression. Here, S k {\displaystyle {\mathfrak {S}}_{k}}

576-736: The first known case of 27 primes (sequence A327760 in the OEIS ): Many of the extensions of Szemerédi's theorem hold for the primes as well. Independently, Tao and Ziegler and Cook, Magyar, and Titichetrakun derived a multidimensional generalization of the Green–Tao theorem. The Tao–Ziegler proof was also simplified by Fox and Zhao. In 2006, Tao and Ziegler extended the Green–Tao theorem to cover polynomial progressions. More precisely, given any integer-valued polynomials P 1 , ..., P k in one unknown m all with constant term 0, there are infinitely many integers x , m such that x  +  P 1 ( m ), ..., x  +  P k ( m ) are simultaneously prime. The special case when

608-528: The largest known Generalized Fermat prime to date, 1963736 + 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. As of 13 December 2022 , PrimeGrid has eliminated 18 values of k from the Riesel problem and is continuing the search to eliminate the 43 remaining numbers. 3 values of k are found by independent searchers. Primegrid worked with

640-587: The largest known generalized Woodall prime, 563528 × 13 − 1 . PrimeGrid's author Rytis Slatkevičius has been featured as a young entrepreneur in The Economist . 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. In the first Citizen Cyberscience Summit , Rytis Slatkevičius gave

672-439: The notation for binomial coefficients , which count the number of k {\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory , the notation [ A ] k {\displaystyle [A]^{k}} is also common, especially when k {\displaystyle k} is a transfinite cardinal number . PrimeGrid PrimeGrid

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704-492: The polynomials are m , 2 m , ..., km implies the previous result that there are length k arithmetic progressions of primes. Tao proved an analogue of the Green–Tao theorem for the Gaussian primes . Subset In mathematics, a set A is a subset of a set B if all elements of A are also elements of B ; B is then a superset of A . It is possible for A and B to be equal; if they are unequal, then A

736-1000: The power set P ⁡ ( S ) {\displaystyle \operatorname {\mathcal {P}} (S)} of a set S , the inclusion partial order is—up to an order isomorphism —the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S ) copies of the partial order on { 0 , 1 } {\displaystyle \{0,1\}} for which 0 < 1. {\displaystyle 0<1.} This can be illustrated by enumerating S = { s 1 , s 2 , … , s k } , {\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},} , and associating with each subset T ⊆ S {\displaystyle T\subseteq S} (i.e., each element of 2 S {\displaystyle 2^{S}} )

768-566: 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,

800-425: The primes. The Green–Tao paper states 'At the time of writing the longest known arithmetic progression of primes is of length 23, and was found in 2004 by Markus Frind, Paul Underwood, and Paul Jobling: 56211383760397 + 44546738095860 ·  k ; k = 0, 1, . . ., 22.'. On January 18, 2007, Jarosław Wróblewski found the first known case of 24 primes in arithmetic progression : The constant 223,092,870 here

832-569: 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: 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 · 2 + 1 at over 2 million digits. On 24 September 2022, PrimeGrid discovered

864-736: The same meaning as and instead of the symbols ⊊ {\displaystyle \subsetneq } and ⊋ . {\displaystyle \supsetneq .} This usage makes ⊆ {\displaystyle \subseteq } and ⊂ {\displaystyle \subset } analogous to the inequality symbols ≤ {\displaystyle \leq } and < . {\displaystyle <.} For example, if x ≤ y , {\displaystyle x\leq y,} then x may or may not equal y , but if x < y , {\displaystyle x<y,} then x definitely does not equal y , and

896-442: The search for a Sophie Germain prime yielded a new record twin prime in September 2016 upon finding the number 2996863034895 × 2 ± 1 composed of 388,342 digits. As of 22 April 2018 , the project has discovered the four largest Woodall primes known to date. The largest of these is 17016602 × 2 − 1 and was found in 21 March 2018. The search continues for an even bigger Woodall prime. PrimeGrid also found

928-464: The search up to  n  = 25 M . Primes known for 3 · 2  + 1 occur at the following n : Primes known for 3 · 2  − 1 occur at the following n : 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: Next target of

960-917: The statement A ⊆ B {\displaystyle A\subseteq B} by applying a proof technique known as the element argument : Let sets A and B be given. To prove that A ⊆ B , {\displaystyle A\subseteq B,} The validity of this technique can be seen as a consequence of universal generalization : the technique shows ( c ∈ A ) ⇒ ( c ∈ B ) {\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen element c . Universal generalisation then implies ∀ x ( x ∈ A ⇒ x ∈ B ) , {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which

992-533: The subproject Seventeen or Bust (for solving the Sierpinski problem ) was added. The calculations for the Riesel problem followed in March 2010. As of January 2023 , PrimeGrid is working on or has worked on the following projects: 321 Prime Search is a continuation of Paul Underwood's 321 Search which looked for primes of the form 3 · 2  − 1. PrimeGrid added the +1 form and continues

Green–Tao theorem - Misplaced Pages Continue

1024-595: 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

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