21. The Prime Glossary: Twin Prime Conjecture There is also a strong form of this conjecture HL23 which states that there areabout. twin primes less than or equal to x. The constant written above as an http://primes.utm.edu/glossary/page.php?sort=TwinPrimeConjecture

22. Twin Primes -- From MathWorld (8). Proof of this conjecture would also imply the existence an infinitenumber of twin primes. Some large twin primes are , , and . http://mathworld.wolfram.com/TwinPrimes.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Mathematical Problems Unsolved Problems ... Sondow Twin Primes Twin primes are pairs of primes of the form p ). The term "twin prime" was coined by Paul Stäckel (1892-1919; Tietze 1965, p. 19). The first few twin primes are for n = 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (Sloane's ). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (Sloane's and It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture ), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem , which states that the number obtained by adding the reciprocals of the odd twin primes converges to a definite number ("

23. Enumeration To 1.6*10^15 Of The Twin Primes And Brun's Constant of K_2 remains undecidedthe famous "twin-primes conjecture " a topic of discussion even in a recent than the twins. The twin-primes conjecture is a consequence of a http://www.trnicely.net/twins/twins2.html

Enumeration to 1.6*10^15 of the twin primes and Brun's constant Thomas R. Nicely http://www.trnicely.net Current e-mail address Date of first release 16 July 1999. Last revised 0357 GMT 12 March 2004. The content of this document (other than the addendum, which was not part of the submission for publication) is essentially that of the original release, except that information rendered obsolete by subsequent events has been removed or modified, in both the main document and the addendum (this liberty is taken in view of the fact that the paper was never accepted for publication). There may also be differences in formatting, and in minor details and corrections. Abstract Enumeration of the twin primes, and the sum of their reciprocals, is extended to 1.6*10^15. An improved estimate is obtained for Brun's constant, B_2 = 1.90216 05824 +/- 0.00000 00030. Error analysis is presented to support the contention that the stated error bound represents a 99 % confidence level. Mathematics Subject Classification 2000 (MSC2000) Primary: 11A41.

24. A New Error Analysis For Brun's Constant Enumeration to 10^14 of the twin primes and Brun's constant. The count pi_2(x) of the number of twinprime pairs = x, as well as the count pi(x) of the number of primes = x, is tabulated by unexpected when dealing with Brun's constant, the twin-primes conjecture, and the Hardy-Littlewood approximation, all of http://www.trnicely.net/twins/twins.html

Enumeration to 10^14 of the twin primes and Brun's constant Thomas R. Nicely http://www.trnicely.net Current e-mail address Document History Last modified....................0355 GMT 12 March 2004. Mathematical Reviews.............MR 97e:11014. Journal citation.................Virginia Journal of Science 46:3 (Fall, 1995) 195-204. Printing date....................March 1996. Accepted for publication.........15 November 1995. Acknowledgment of submission.....31 August 1995. Original submission..............28 August 1995. Except for the addendum (not part of the submission for publication), the content of this document is essentially that of the journal article as published; it may contain minor revisions and corrections, and differ in format and detail. Abstract Mathematics Subject Classification 2000 (MSC2000) Primary: 11A41. Secondary: 11-04, 11N36, 11Y60, 11Y70, 11Y35, 68M15. Key words and phrases Twin primes, Brun's constant, primes, P*ntium flaw, computer errors.

25. Twin Prime Conjecture - Reference Library a generalization of the twin Prime conjecture, known as the Hardy Littlewood conjecture,which is concerned with the distribution of twin primes, in analogy http://www.campusprogram.com/reference/en/wikipedia/t/tw/twin_prime_conjecture_1

Reference Library: Encyclopedia Main Page See live article Alphabetical index Twin prime conjecture The twin prime conjecture is a famous unsolved problem in number theory that involves prime numbers . It states: There are an infinite number of primes p such that p + 2 is also prime. Such a pair of prime numbers is called a twin prime . The conjecture has been researched by many number theorists. The majority of mathematicians believe that the conjecture is true, based on numerical evidence and heuristic reasoning involving the probabilistic distribution of primes. In de Polignac made the more general conjecture that for every natural number k , there are infinitely many prime pairs which have a distance of 2 k . The case k =1 is the twin prime conjecture. Partial results In showed that there is a constant c < 1 and infinitely many primes p such that p p c ln p ), where p ' denotes the next prime after p . This result was successively improved; in Maier showed that a constant c < 0.25 can be used. In , Jing-run Chen showed that there are infinitely many primes p such that p +2 is a product of at most two prime factors. The approach he took involved a topic called Sieve theory, and he managed to treat the Twin Prime Conjecture and

26. Twin Prime Conjecture - Encyclopedia Article About Twin Prime Conjecture. Free A A strong form of the twin Prime conjecture, the HardyLittlewood conjecture, postulatesa distribution law for twin primes akin to the prime number theorem. http://encyclopedia.thefreedictionary.com/Twin Prime Conjecture

Dictionaries: General Computing Medical Legal Encyclopedia Twin Prime Conjecture Word: Word Starts with Ends with Definition The twin prime conjecture is a famous unsolved problem in number theory Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers. Number theory may be subdivided into several fields according to the methods used and the questions investigated. See for example the list of number theory topics. Click the link for more information. that involves prime numbers In mathematics, a prime number , or prime for short, is a natural number greater than 1 whose only positive divisors are 1 and itself. The sequence of prime numbers (sequence A000040 in OEIS) begins See list of prime numbers for the first 500 primes. Click the link for more information. . It states: There are an infinite number of primes p such that p + 2 is also prime.

27. Twin Prime Conjecture From MathWorld twin Prime conjecture from MathWorld There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/TwinPrimeConjectur

28. Introduction To Twin Primes And Brun's Constant Computation 2.2 twin prime conjecture. R d, 1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3. According to thisconjecture the density of twin primes is equivalent to the density of cousin primes. http://numbers.computation.free.fr/Constants/Primes/twin.html

Introduction to twin primes and Brun's constant computation (Click here for a Postscript version of this page and here for a pdf version) Introduction It's a very old fact (Euclid 325-265 B.C., in Book IX of the Elements ) that the set of primes is infinite and a much more recent and famous result (by Jacques Hadamard (1865-1963) and Charles-Jean de la Vallee Poussin (1866-1962)) that the density of primes is ruled by the law p (n) n log(n) where the prime counting function p (n) is the number of prime numbers less than a given integer n. This result proved in 1896 is the celebrated prime numbers theorem and was conjectured earlier, in 1792, by young Carl Friedrich Gauss (1777-1855) and by Adrien-Marie Legendre (1752-1833) who studied the repartition of those numbers in published tables of primes. This approximation may be usefully replaced by the more accurate logarithmic integral Li(n): p (n) Li(n)= n dt log(t) However among the deeply studied set of primes there is a famous and fascinating subset for which very little is known and has generated some famous conjectures: the twin primes (the term prime pairs was used before [ Definition 1 A couple of primes (p,q) are said to be twins if q=p+2. Except for the couple (2,3), this is clearly the smallest possible distance between two primes.

29. Twin Prime Conjecture Proof The six wide array further helps to demonstrate the otherwise still unproven conjecturethat there must be infinitely many twin primes, that is, pairs of http://www.recoveredscience.com/primes1ebook02.htm

recoveredscience .com We offer surprises about in our e-book Prime Passages to Paradise by H. PeterAleff Site Contents PRIME PATTERNS Table of Contents Rectangular arrays Twin prime proof Prime facts Prime problems Polygonal numbers Number pyramids ... Reader responses Visit our Sections: Constants Board Games Astronomy Medicine Store Stuff Home Page Search this site FAQ about e-books Download free e-books ... email us Footnotes : Paulo Ribenboim: "The New Book of Prime Number Records", Springer- Verlag, New York, 1996, pages 20 to 52. Calvin C. Clawson: "Mathematical Mysteries: The Beauty and Magic of Numbers", Plenum Press, New York, 1996, pages 64 and 65 Volume 1: Patterns of prime distribution in "polygonal - number pyramids" You are on page Twin Prime Proof (To German translation - zur deutschen Übersetzung) 1.2. Proving the twin prime conjecture The six- wide array further helps to demonstrate the otherwise still unproven conjecture that there must be infinitely many twin primes , that is, pairs of numbers where p and p + 2 are both prime. Here is how:

30. Primes Index One of these organizing arrangements allows a simple proof for the long standingtwin prime conjecture that the quantity of twin primes, or pairs of primes http://www.recoveredscience.com/primes_index.htm

recoveredscience .com We offer surprises about in our e-book Prime Passages to Paradise by H. PeterAleff Site Contents PRIME PATTERNS Table of Contents Rectangular arrays Twin prime proof Prime facts ... Reader responses Visit our Sections: Constants Board Games Astronomy Medicine Store Stuff Home Page Search this site FAQ about e-books Download free e-books ... email us Prime number alignments long forgotten but long ago vital An exploration of patterns in the distribution of primes and some evidence for their ancient perception The first volume in this series discusses different ways of organizing the natural numbers to make them reveal some of the order in the distribution of primes . Primes are numbers which have no factors except one and themselves and they are the basic building blocks from which all other numbers are made One of these organizing arrangements allows a simple proof for the long- standing twin prime conjecture that the quantity of twin primes, or pairs of primes which are separated only by the even entry between them, is infinite. You will further find here descriptions and illustrations of some surprising patterns formed by primes These visually striking patterns appear already in the first few thousand entries when you write the consecutive natural numbers in simple triangle- shaped arrays where the ends of the successive layers are, for instance, the successive squares 1, 4, 9, 16, 25, and so on, and you then mark the primes.

31. Mathenomicon.net : Reference : Twin Prime Conjecture The unproven conjecture that there are an infinite number of twin primes thatis, an infinite number of integers such that and are both prime numbers. http://www.cenius.net/refer/display.php?ArticleID=twinprimeconjecture

32. Twin Prime Conjecture twin Prime conjecture. twin Prime conjecture. The twin Prime conjecturestates that there are an infinite number of twin primes. A http://www.users.globalnet.co.uk/~perry/maths/twinprimeconjecture/twinprimeconje

Twin Prime conjecture [Home] [Other Maths] Twin Prime conjecture The Twin Prime conjecture states that there are an infinite number of twin primes. A twin prime is defined as a pair of numbers, 6k-1 and 6k+1, such that both are prime. Proof i.e. the TPC is equivalent to the conjecture that there are an infinite number of integers with only even anti-divisors. As 3 as an anti-divisor leaves only multiples of 3 as a candidate, then we only need consider prime anti-divisors greater than or equal to 5. We only need consider prime anti-divisors, as numbers with odd composite anti-divisors also have the prime factors of these composites as anti-divisors. A number with an odd anti-divisor can be written as kp+(p-1)/2 or as kp+(p+1)/2. But we only need to consider integers 0mod3, and this allows us to eliminate some possibilities. To do this, consider the two forms of primes, 6k-1 and 6k+1. Note that these are not twin primes, but that all primes after 3 are of one of these forms. If we look at 6k-1, then the integers we can create are j(6k-1) + 3k - 1 and j(6k-1) + 3k. In both of these cases, if j=1mod3, then neither are divisible by 3, and so we can ignore these possibilites.

33. Page 012 Introduction to twin primes Link . The twin prime conjecture Link . Daniel Zwillinger,A Goldbach conjecture using twin primes, Math. Comp. 33 (1979), no. http://www.math.utoledo.edu/~jevard/Page012.htm

Twin primes and their applications Page maintained by Jean-Claude Evard. Last update: June 17, 2003. AMS classification numbers Link Link Link Link ... Link Copies of reviews from Mathematical Review cannot be posted on Web pages, but they can be seen through links to MathSciNet. These links work only in the networks of institutions or on the computers of users who are current subscribers to MathSciNet. Main directions of research about twin primes 1. The twin prime conjecture and extensions of Chen's theorem Link 2. Search for lower and upper bounds for the number of twin primes in intervals [1, n]. 3. Search for the largest known twin primes. 4. Computation of Brun's constant. 5. Computation of the twin prime constant. 6. Generalizations to Gaussian primes. 7. Applications of twin primes. Twin primes on large Web sites . List of 114 Web sites about twin primes, with summaries Link provided by Geometry, The Online Learning Center. . Web pages related to twin primes maintained by Eric Weisstein Link on the Web site MathWorld Link at Wolfram Research Link Introduction to twin primes Link The twin prime conjecture Link The twin prime constant Link The Brun's constant Link 3. Web pages related to twin primes

34. Prime Finding: Mathematicians Mind The Gap: Science News Online, March 29, 2003 The twinprimes conjecture, one of the most famous unsolved problems in number theory,speculates that there are infinitely many pairs of primes that differ by http://www.sciencenews.org/20030329/fob1.asp

Math Trek Crediting Basketball's Three-Pointers Food for Thought Cranberry Juice—A Cocktail for the Heart Science Safari Home Base for Government Science TimeLine 70 Years Ago in Science News Week of March 29, 2003; Vol. 163, No. 13 , p. 195 Prime Finding: Mathematicians mind the gap Erica Klarreich A mathematical duo has made a surprising advance in understanding the distribution of prime numbers, those whole numbers divisible only by themselves and 1. The new result is the most exciting work on prime numbers in more than 3 decades, says mathematician Hugh L. Montgomery of the University of Michigan in Ann Arbor. However, he cautions that experts are still checking the details of the proof. Among small numbers, primes are common. Of the first 10 numbers, for instance, 4 of them—2, 3, 5, and 7—are prime. But among larger numbers, primes thin out. Around a trillion, for instance, only about 1 in every 28 numbers is prime. In the late 19th century, mathematicians proved that the distribution of primes follows an amazingly simple pattern: The average spacing between primes near a number x is the natural logarithm of x, a number closely related to the number of digits in x. This formula is true only on average, however. Sometimes, the gap between primes is much smaller, other times much larger. The twin-primes conjecture, one of the most famous unsolved problems in number theory, speculates that there are infinitely many pairs of primes that differ by only two. Examples of twin-primes abound—17 and 19, for instance—but for more than a century, mathematicians have struggled without success to prove the conjecture.

35. Prime Conjectures And Open Question conjectured by Polignac 1849. When n=1 this is the twin prime conjecture. twinPrime conjecture There are infinitely many twin primes. http://www.utm.edu/research/primes/notes/conjectures/

Prime Conjectures and Open Questions (Another of the Prime Pages ' resources) Home Search Site Largest The 5000 ... Submit primes Below are just a few of the many conjectures concerning primes. Goldbach's Conjecture: Every even n Goldbach wrote a letter to Euler in 1742 suggesting that . Euler replied that this is equivalent to this is now know as Goldbach's conjecture. Schnizel showed that Goldbach's conjecture is equivalent to distinct primes It has been proven that every even integer is the sum of at most six primes [ ] (Goldbach's conjecture suggests two) and in 1966 Chen proved every sufficiently large even integers is the sum of a prime plus a number with no more than two prime factors (a P ). In 1993 Sinisalo verified Goldbach's conjecture for all integers less than 4 ]. More recently Jean-Marc Deshouillers, Yannick Saouter and Herman te Riele have verified this up to 10 with the help, of a Cray C90 and various workstations. In July 1998, Joerg Richstein completed a verification to 4

36. There Are Infinitely Many Primes, But, How Big Of An Infinity? 1567). On the other hand, it is conjectured that there are infinitely many twinprimes (this is the twin prime conjecture, most everybody believes it is true http://www.utm.edu/research/primes/infinity.shtml

There are infinitely many primes, but, how big of an infinity? (Another of the Prime Pages ' resources) Home Search Site Largest The 5000 ... Submit primes Contents: Introduction - What is the question? Countability (is there another way to distinguish infinite sets?) Convergence and Divergence (definitions and examples) The Sum of the Reciprocal of the Primes (diverges! but for the twin primes...) 1. Introduction - What is the Question? About 2000 years ago Euclid proved that there were infinitely many primes. For mathematicians "infinity many" is an incomplete answerthey then ask "how big of an infinity?" The prime number theorem, which states the number of primes less than x is approximately x /log x (the natural log), gives perhaps the best answer. Another way to answer that question is to ask whether or not the sum of the inverses of the primes convergesthat is, what happens when we add up the following fractions? If you understand the terms convergence and divergence , then jump to section four below.

37. Prime Strings, Goldbach And His Evil Twin For now, I want to talk about the Goldbach conjecture and twin primes. We willjust pretend that I already have a binary string representing the primes. http://descmath.com/prime/prime_strings.html

Prime Strings, Goldbach and his Evil Twin The two most famous mathematical conjectures concerning primes are: The Twin Prime Conjecture and the Goldbach Conjecture. The Twin Prime Conjecture speculates that there is an infinite number of primes pairs p1 and p2 such that (p2 - p1 = 2). The Goldbach Conjecture stipulates that all even numbers can be written as the sum of two primes. I am inclined to believe that both conjectures are true. But, like most proofs that involve establishing a truth for an infinite collection, the postulates are devilishly difficult to prove. I've found that representing the primes with a binary string, makes it is easy to see the relation between the Goldbach and Twin Prime conjectures. A binary string is simply a long string of boolean characters. A boolean character has only two possible values. The boolean value is either on or off, true or false. Computer programmers often express binary strings as a series of 1s and 0s. To represent the primes as a binary string, I simply look at each number starting with 1. If the number is prime, I record a "1". If not, I record "0". The nth value in this binary string will be 1 if n is prime, else 0. When discussing the Twin Prime and Goldbach conjectures, I find it easiest to drop the even numbers. To create a binary string that represents the odd integers, I start with a list of dds then record if it is prime:

38. Prime Territory If you can show that some of the twin primes survive subsequent iterationsof the sieve, then you would prove the conjecture. To http://descmath.com/prime/pp.html

Prime Patterns The title "prime pattern" is a misnomer. Prime numbers, by their nature, have no pattern. A prime number is simply a number that is not a composite. A prime number is defined as a positive integer that is divisible only by itself and one. The set of prime numbers is what remains after you remove the composites from the set of natural numbers. Primes are more of an anti-pattern than a pattern. I made the illusion earlier that prime numbers are transcendental. Mathematicians use the word transcendental to refer to numbers like pi and Euler's constant. Like irrational numbers, transcendental numbers cannot be expressed as a ratio between two integers. Even more amazing, transcendental numbers cannot be expressed with a finite algebraic equation. Now, I suspect that the prime numbers have this same behavior. I doubt anyone will ever find a finite algebraic equation to generate the primes. Any equation made with a finite number of exponents, additions and multiplications would involve a repetition at some finite level. The primes, however, transcend such repetitious behavior. To create a list primes, we create a program called a sieve. Sieves calculate composites. We assume the remainder to be prime. Prime sieves are self-referential. A prime sieve needs to look at the previous iterations of the sieve to determine which numbers are prime. Prime sieve programs often include recursive loops.

39. WANLESS THEOREM WANLESS THEOREM. There exist an infinite number of pairs of primes, P and Q, stPQ=2N, for all N (Corollaries include affirmation of the twin Prime conjecture). http://www.bearnol.pwp.blueyonder.co.uk/Math/wanless2.html

WANLESS' THEOREM There exist an infinite number of pairs of primes, P and Q, s.t. P-Q=2N, for all N (Corollaries include affirmation of the Twin Prime Conjecture) Proof: Let p0,p1,p2... be the positive primes, including 1, in inceasing order. Let Pn=p0p1p2...p(i0-1)p(i0+1)...p(i1-1)p(i1+1)...pn + (p(i0)**j0)(p(i1)**j1)... and let Qn=p0p1p2...p(i0-1)p(i0+1)...p(i1-1)p(i1+1)...pn - (p(i0)**j0)(p(i1)**j1)... [any i, j] Note that hcf (Pn, p0p1p2p3p4...pn) = 1 and hcf (Qn, p0p1p2p3p4...pn) = 1 [Euclid] Then Pn and Qn are either both prime or Pn is divisible by a prime greater than pn or Qn is divisible by a prime greater than pn Corollary (Twin Prime Conjecture): (3) [N=1]

40. Twin Primes 135,780,321,665. See also Brun s Constant, de Polignac s conjecture Prime Constellation,Sexy primes, twin Prime conjecture, twin primes Constant. References. http://icl.pku.edu.cn/yujs/MathWorld/math/t/t437.htm

Twin Primes Twin primes are Primes ) such that . The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (Sloane's ). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (Sloane's and Let be the number of twin primes and such that . It is not known if there are an infinite number of such Primes (Shanks 1993), but all twin primes except (3, 5) are of the form . J. R. Chen has shown there exists an Infinite number of Primes such that has at most two factors (Le Lionnais 1983, p. 49). Bruns proved that there exists a computable Integer such that if , then (Ribenboim 1989, p. 201). It has been shown that where has been reduced to (Fouvry and Iwaniec 1983), (Fouvry 1984), 7 (Bombieri et al. 1986), 6.9075 (Fouvry and Grupp 1986), and 6.8354 (Wu 1990). The bound on is further reduced to 6.8325 (Haugland 1999). This calculation involved evaluation of 7-fold integrals and fitting of three different parameters. Hardy and Littlewood conjectured that (Ribenboim 1989, p. 202).

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