21. Prime Numbers The twin primes conjecture that there are infinitely many pairs of primes only2 apart. Goldbach s Conjecture (made in a letter by C Goldbach to Euler in http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html

Prime numbers Number theory index History Topics Index Version for printing Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians. The mathematicians of Pythagoras 's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers. A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28. A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa. You can see more about these numbers in the History topics article Perfect numbers By the time Euclid 's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

22. Fred Holt: Expected Gaps Between Prime Numbers The most famous of these is the Twin Prime Conjecture, that there are an Generalizing the twin primes conjecture is a conjecture by Polignac from 1849 http://www.math.washington.edu/~papazoga/cpm/2005/Winter/holt.html

Current Problems in Mathematics Speaker Fred Holt Title Expected gaps between prime numbers Date January 13 4:00pm PDL C-36

23. John Quiggin » Blog Archive » What I’m Reading, And More for the best generalization of the twin primes conjecture, due to Hardy andLittlewood. So I think RH is in a class above the twin primes conjecture. http://johnquiggin.com/index.php/archives/2003/07/13/what-im-reading-and-more/

John Quiggin Producers and consumers Libertarianism, continued by HM A Beautiful Mind Meanwhile, Sunday being the day of religious observance in Australia, I finally did something about the change of religion La Marseillaise .This set me thinking about other possibilities -perhaps the Horst Wessel Lieder The Star-Spangled Banner This was the first AFL SANFL 40 This entry was posted on Sunday, July 13th, 2003 at 8:14 pm and is filed under Books and culture . You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed. Brian Weatherson Says: July 14th, 2003 at 10:01 am John Quiggin Says: July 14th, 2003 at 11:10 am Scott Wickstein Says: July 15th, 2003 at 1:00 am The Horst Weseel Lied for Carlton? OUCH The reason why your ticket prices are a tad higher is they pay the players properly these days. All things are relative. British blogger Stephen Pollard said he pays 750 for a season ticket to English Premier League club Spurs. That works out to over $100 a game, per seat.

24. Sci.math FAQ: Unsolved Problems Collatz Problem * Goldbach s conjecture * twin primes conjecture_ Names of largenumbers http://www.faqs.org/faqs/sci-math-faq/unsolvedproblems/

Usenet FAQs Search Web FAQs Documents ... RFC Index sci.math FAQ: Unsolved Problems There are reader questions on this topic! Help others by sharing your knowledge Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca hv@cix.compulink.co.uk (Hugo van der Sanden): To the best of my knowledge, the House of Commons decided to adopt the US definition of billion quite a while ago - around 1970? - since which it has been official government policy. dik@cwi.nl (Dik T. Winter): The interesting thing about all this is that originally the French used billion to indicate 10^9, while much of the remainder of Europe used billion to indicate 10^12. I think the Americans have their usage from the French. And the French switched to common European usage in 1948. gonzo@ing.puc.cl alopez-o@barrow.uwaterloo.ca Rate this FAQ N/A Worst Weak OK Good Great Related questions and answers Usenet FAQs Search Web FAQs ... RFC Index Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update June 15 2004 @ 00:31 AM

25. Twin Primes -- From MathWorld It is conjectured that there are an infinite number of twin primes (this is This result is sometimes called the strong twin prime conjecture and is a http://mathworld.wolfram.com/TwinPrimes.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Prime Numbers Prime Clusters ... Sondow Twin Primes Twin primes are pairs of primes of the form ). The term "twin prime" was coined by Paul Stäckel (1892-1919; Tietze 1965, p. 19). 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 All twin primes except (3, 5) are of the form 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 ("

26. The Prime Glossary: Twin Prime Conjecture Welcome to the Prime Glossary a collection of definitions, information and factsall related to prime numbers. This pages contains the entry titled twin http://primes.utm.edu/glossary/page.php?sort=TwinPrimeConjecture

27. Twin Prime Conjecture - Definition Of Twin Prime Conjecture In Encyclopedia The twin prime conjecture is a famous problem in number theory that involvesprime numbers. It statesThere are an infinite number of primes p such that p + http://encyclopedia.laborlawtalk.com/Twin_Prime_Conjecture

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law The twin prime conjecture is a famous 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. Mathematicians believe the conjecture to be true, based only 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. Contents showTocToggle("show","hide") 1 Partial results 2 Hardy-Littlewood conjecture 3 Serious problem found in potential proof 1 See also ... 2 External links Partial results In Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous result was the first use of the Brun sieve and helped initiate the development of modern sieve theory . The modern version of Brun's argument can be used to show that the number of twin primes less than N does not exceed C N / log N for some absolute constant In showed that there is a constant c p such that p p c ln p , where p ' denotes the next prime after p . This result was successively improved; in

28. Twin Prime - Definition Of Twin Prime In Encyclopedia A strong form of the twin Prime conjecture, the HardyLittlewood conjecture,postulates a distribution law for twin primes akin to the prime number theorem. http://encyclopedia.laborlawtalk.com/Twin_prime

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law A twin prime is a prime number that differs from another prime number by two . Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are 5 and 7, 11 and 13, and 821 and 823. (Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the Twin Prime Conjecture . A strong form of the Twin Prime Conjecture, the Hardy-Littlewood conjecture , postulates a distribution law for twin primes akin to the prime number theorem Using his celebrated sieve method Viggo Brun . This result implies that the sum of the reciprocals of all twin primes converges (see Brun's constant ). This is in stark contrast to the sum of the reciprocals of all primes, which diverges. He also shows that every even number can be represented in infinitely many ways as a difference of two numbers both having at most 9 prime factors. Chen Jingrun 's well known theorem states that for any m even, there are infinitely many primes that differs by m from a number having at most two prime factors. (Before Brun attacked the twin prime problem

29. 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, 6k1 and 6k+1, 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 (prime) 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.

30. Introduction To Twin Primes And Brun's Constant Computation conjecture 3 twin prime conjectureFor large values of n, According to thisconjecture the density of twin primes is equivalent to the density of 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.

31. Conjecture 20. The First N Natural Numbers Listed In An Order Such That The Sum has solved completely this conjecture from an unexpected way that involvesthe twin primes. Where P is the larger value from two twin primes. http://www.primepuzzles.net/conjectures/conj_020.htm

Conjectures Conjecture The first N natural numbers listed in an order such that the sum of each two adjacent of them is a prime number, and the Rivera's Algorithm Example: for N=10 one solution is Needless to say that for each N value there are several solutions that differs between them by the order that the numbers are listed in the row (*).Other thing that should be said is that for the purposes of this puzzle, the (N-1) primes obtained need not to be distinct. But I went not interested in the question about the quantity of distinct solutions for each N. Instead of that, I asked my self if it could exist a for getting at least one solution for each and any N. What I wanted to discover was a method for completing the assignation of the N values in exactly N safe steps, without having to go back, erase and start again when the remaining values to assign exhibit no one available option. What is that of "

32. Prime Territory distribution of the primes, the Goldbach conjecture and the twin prime conjectures . The junction between patterns creates a twin prime candidate. 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.

33. Mathenomicon.net : Reference : Twin Prime Conjecture Definition of twin prime conjecture. The unproven conjecture that there arean infinite number of twin primes that is, an infinite number of integers http://www.cenius.net/refer/display.php?ArticleID=twinprimeconjecture

34. Mathenomicon.net : News : A Possible Proof Of The Twin Prime Conjecture University appears to have found a proof of the twin prime conjecture.twin primes are pairs of prime numbers $ (p,p+2)$ . The conjecture states that http://www.cenius.net/news/news.php?ArticleID=12

35. Twin Prime Conjecture Proof Proving the twin prime conjecture. The six wide array further helps to still unproven conjecture that there must be infinitely many twin primes, 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:

36. MathForge.netPower Tools For Online Mathematics 71 and 73 are examples of twin primes. The twin Prime conjecture posits thatthere are infinitely many twin primes. Until now, there has been no proof. http://mathforge.net/index.jsp?page=seeReplies&messageNum=589

37. The Music Of Primes This is exactly what has yet to be proven in the twin Prime conjecture. The conjecture says that you will always find clusters of primes where N and N+2 http://www.musicoftheprimes.com/films.htm

The art of maths Maths and the creative arts :: Films back to creative arts index Contact contact-themovie.warnerbros.com Pi Max Cohen, the hero of this film, is a mathematician who spends his life tracking patterns in the decimal expansion of pi. Convinced these numbers hold the key to the stock market and the Jewish kabbalah, he eventually goes crazy. www.pithemovie.com top A beautiful mind http://www.abeautifulmind.com top The cube Six characters wake up inside a complicated system of interconnected cubes. Some of the rooms are trapped. They soon discover that if the number of the room is a prime number then the room contains a deadly trap. The primes are the key to their survival [link to prime number cicadas] http://www.cubethemovie.com

38. Enumeration To 1.6*10^15 Of The Twin Primes And Brun's Constant The twinprimes conjecture is a consequence of a much stronger result proposedby Hardy and Littlewood 9 in 1923, 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 0815 GMT 13 September 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.

39. FOM: Re: Twin Primes Again The reason I used the twin prime conjecture rather than Goldbach s conjecture (Fermat sconjecture has been proven) is that the independence of the TPC from http://www.cs.nyu.edu/pipermail/fom/2000-June/004123.html

FOM: Re: twin primes again Joe Shipman shipman at savera.com Wed Jun 21 12:40:36 EDT 2000 Previous message: FOM: twin primes again Next message: FOM: Re: twin primes again Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] I understand from your contributions that the twin prime conjecture is something different from Goldbach's conjecture or Fermat's last theorem. Do I correctly understand that, according to your opinion, no position is possible which simultaneously (a) does not assume that the truth-value of such "highly infinitary" statements as the twin prime conjecture is determinated from the outset; (b) does not deny the whole set of integers as a "completed whole", as something "to quantify over"; (c) does not distinguish between statements like "for each integer ..." and the corresponding "universally quantified" formula? Note that (a) is a crucial point for every constructive philosophy, if not for any pragmatic view of mathematics in general; (b) is just what I tend to assign to (Bishop's) constructive mathematics, although Bishop possibly would not agree;

40. FOM: Twin Primes Vs. Goldbach Conjecture FOM twin primes vs. Goldbach conjecture Couldn t also the falsehood of thereare infinitely many twin primes be finitely veryfied by exhibiting the http://www.cs.nyu.edu/pipermail/fom/2000-June/004111.html

FOM: twin primes vs. Goldbach conjecture Peter Schuster pschust at rz.mathematik.uni-muenchen.de Mon Jun 19 10:30:04 EDT 2000 Previous message: FOM: Re: science and constructive mathematics Next message: FOM: Some thought on "Realism" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] The problem with using Goldbach's conjecture as an example of a possibly indeterminate statement is that it is hard to imagine how it could be both false and unknowable, because a counterexample can be finitely verified. This asymmetry obscures the relationship between "unknowable" and "indeterminate" that I was trying to illustrate. Couldn't also the falsehood of "there are infinitely many twin primes" be finitely veryfied by exhibiting the greatest pair and by giving a proof that it is so? Such a proof might even be simpler than all the calculations necessary for demonstrating that some large even integer is not sum of two prime numbers. Peter Schuster. Previous message: FOM: Re: science and constructive mathematics Next message: FOM: Some thought on "Realism"

Page 2     21-40 of 94    Back | 1  | 2  | 3  | 4  | 5  | Next 20