41. Web Resources Guide For Chapter 3 sieve of Eratosthenes and for exploring the distribution of prime numbers.Information about twin primes and the Goldbach conjecture is also provided. http://www.mhhe.com/math/advmath/rosen/r5/student/ch03/weblinks.html

Information Center Instructor Center Student Center Home Student Center Interactive Demos Self Assessments A Guide to Writing Proofs Common Mistakes in Discrete Math ... Bulletin Board Chapter Level Resources Choose one ... Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Web Resources Guide Maple Supplement Extra Examples Extra Steps Discrete Mathematics and Its Applications Web Resources Guide for Chapter 3 Page 214 Guidance on how to write proofs as well as an introduction to proof strategies, can be found at Larry W. Cusick's website. Go to How To Write Proofs: http://zimmer.csufresno.edu/~larryc/proofs/proofs.html Another website with more information on several different methods of proof can be found at Proving Real Theorems: http://cse.stanford.edu/classes/cs103a/h24RProofs.htm An interesting variety of proofs and other problems in mathematics can be found at the Cut-the-Knot site. Proofs in Mathematics: http://www.cut-the-knot.com/proofs/index.shtml A selection of proofs by contradiction can be found at Proofs by Contradiction: http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html

42. Science -- Sign In No one knows whether twin primes ever stop appearing. The new proof is still afar cry from the twin prime conjecture, but it offers a glimmer of hope that http://www.sciencemag.org/cgi/content/full/308/5726/1238

You do not have access to this item: Full Text : Cipra, NUMBER THEORY: Third Time Proves Charm for Prime-Gap Theorem, Science You are on the site via Free Public Access. What content can I view with Free Public Access If you have a personal user name and password, please login below. SCIENCE Online Sign In Options For Viewing This Content User Name Password this computer. Help with Sign In If you don't use cookies, sign in here Join AAAS and subscribe to Science for free full access. Sign Up More Info Register for Free Partial Access including abstracts, summaries and special registered free full text content. Register More Info Pay per Article 24 hours for US $10.00 from your current computer Regain Access to a recent Pay per Article purchase Need More Help? Can't get past this page? Forgotten your user name or password? AAAS Members activate your FREE Subscription

43. ::MHC Math Club: PROBLEM Of The WEEK:: Note This problem is related to the twin prime conjecture. This is the conjecturethat there are infinitely many pairs of twin primes. twin primes are http://www.mtholyoke.edu/~ysasanne/math/mathclubproblem.html

Problem of the Week #8 (November 29) Given a segment AB in the plane and a line L in the same plane, let n(L) be the number of points C on L such that triangle ABC is isosceles. For example, I L is the perpendicular bisector of AB, then n(L)= infinity. What are the other possible values of n(L)? And how do they occur? Solutions should be placed in Harriet Pollatsek's mail slot in the department office or slid under her door (400 Clapp) by 5:00 p.m. Friday, December 3 #7 (November 15) This problem is essentially the Car Talk "Puzzler" from 11/6/04. You are a sales representative for a company based in Delaware. Your boss gives you a new assignment: you have to drive to the company's office in each of the 48 contiguous states, beginning from Delaware. The catch is that you can only enter each state once. While you are sitting at your desk trying to figure out how to do this, the company CEO stops by your cubicle. "I just heard about your new assignment," she says. "You'll be ending up in my home state. I've been looking for an excuse to visit my family there, so I'll meet you there and fly you back to Delaware on my private jet." What state is the CEO from? How did you figure it out? Note: There is a map of the contiguous 48 states hanging in the stairwell near the megatherium.

44. Proof Of Golbach S Conjecture And The Twin Prime Conjecture Proof of Golbach s conjecture and the twin prime conjecture He s also proven twin primes, Riemann hypothesis, and of course Fermat s Last Theorem. http://www.physicsforums.com/archive/t-81643_Proof_of_Golbach's_conjecture_and_t

Technology Services Mathematics Number Theory Proof of Golbach's conjecture and the twin prime conjecture keebs - Proof of Golbach's conjecture and the twin prime conjecture I found this on arxiv...is this guy a loon or do the proofs seem reasonable? Proofs (http://www.arxiv.org/ftp/math/papers/0309/0309103.pdf) Discuss Proof of Golbach's conjecture and the twin prime conjecture Here, Free! TenaliRaman - Proof of Golbach's conjecture and the twin prime conjecture Interesting paper! I believe that the author gave more explanation on his formula to his P(N). Right now to me it seems to have been just plucked out of thin air. Anyways, intelligent nitpicks should be left for the more qualified. AI Discuss Proof of Golbach's conjecture and the twin prime conjecture Here, Free! Zurtex - Proof of Golbach's conjecture and the twin prime conjecture This is not a proof, there are multiple mistakes and they have neglected to prove some areas which I believe to be untrue anyway. I need to head off to the shop, I might sit down later and try and demonstrate why this isn't true unless someone else has either done this or in fact shown me to be wrong. Discuss Proof of Golbach's conjecture and the twin prime conjecture Here, Free!

45. Math Trek: Closing The Gap On Twin Primes, Science News Online, July 16, 2005 These socalled twin primes are scattered throughout the list of all prime numbers . a proof of the twin prime conjecture may not be far away. http://www.sciencenews.org/articles/20050716/mathtrek.asp

Science News Books. Subscribe to Science News ... Science News for Kids Math Trek Closing the Gap on Twin Primes Food for Thought Money Matters in Obesity Science Safari Ocean Habitats TimeLine 70 Years Ago in Science News Science News e-LETTER. ... Week of July 16, 2005; Vol. 168, No. 3 Closing the Gap on Twin Primes Ivars Peterson B.C. ) provided a simple proof that the sequence of prime numbers continues forever. A prime is a whole number (other than 1) that's evenly divisible only by itself and 1. This definition leads to the following sequence of numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, and so on. Suppose there is a finite number of primes, Euclid argued. This means there's also a largest prime, n . Multiply all the primes together, then add 1: (2 x 3 x . . . x n ) + 1. The new number is certainly bigger than the largest prime. If the initial assumption is correct, the new number can't be a prime. Otherwise, it would be the largest. Hence, it must be a composite number and divisible by a smaller number. However, because of the way the number was constructed, any known prime, when divided into the new number, leaves a remainder of 1. Therefore, the initial assumption can't be correct, and there can be no largest prime. Primes often occur as pairs of consecutive odd integers: 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. These so-called twin primes are scattered throughout the list of all prime numbers. However, there's no proof yet that there are infinitely many pairs of primes that differ by only 2.

46. Professeur Badih GHUSAYNI The twin prime conjecture states that the number of twin primes is infinite. The objective of this note is to tie the twin prime conjecture to complex http://www.ul.edu.lb/francais/publ/ghus.htm

Professeur Badih GHUSAYNI Name : Dr. Badih Ghusayni Email : bgou@ul.edu.lb badih@futureintech.net Faculty of Science -1, Department of Mathematics, Lebanese University Managing Editor, International Journal of mathematics and computer science http://ijmcs.futureintech.net/index.htm Research Interests and Specialties Complex and Harmonic Analysis : Entire functions and Fourier Transforms, Representation of Entire Functions by Series and Integrals. Approximation Theory : Approximation by a Nonfundamental Sequence of translates Analytic Number Theory : Tauberian Theorems, Distribution of Primes, Twin Primes, Perfect Numbers, The Zeta Function at Odd Arguments, Factorization and Primality. Computerized Instruction : Maple Publications Books Number Theory from an analytic point of view ISBN 9953-0-0282-7 : Badih Ghusayni Paperback 198 pages Contents Overview of Complex Numbers and Functions. Hadamard Factorization Theorem and Entire Functions of Order One and Infinite Type. The Goldbach and Twin Prime Conjectures. Zeta of 3.

47. CJM - Primes In Short Segments Of Arithmetic Progressions Assuming a strong form of the twin prime conjecture and the Riemann Hypothesisone can obtain an asymptotic formula for the total variance in the range when http://www.journals.cms.math.ca/cgi-bin/vault/view/goldston0874

CJM (1998) Vol 50 No 3 / pp. 563-580 Primes in short segments of arithmetic progressions D. A. Goldston and C. Y. Yildirim Abstract TeX format For download Keywords none Category Primary: 11M26 Secondary: none

48. Goldbach Conjecture Verification The Goldbach conjecture is one of the oldest unsolved problems in number theory 1, is the twin primes constant. In 3, Crandall and Pomerance suggest http://www.ieeta.pt/~tos/goldbach.html

Goldbach conjecture verification Introduction News Results Top 20 ... [Up] Introduction The Goldbach conjecture is one of the oldest unsolved problems in number theory [1, problem C1] . In its modern form, it states that every even number larger than two can be expressed as a sum of two prime numbers. Let n be an even number larger than two, and let n=p+q , with p and q prime numbers, , be a Goldbach partition of n . Let r(n) be the number of Goldbach partitions of n . The number of ways of writing n as a sum of two prime numbers, when the order of the two primes is important, is thus R(n)=2r(n) when n/2 is not a prime and is R(n)=2r(n)-1 when n/2 is a prime. The Goldbach conjecture states that , or, equivalently, that , for every even n larger than two. In their famous memoir [2, conjecture A] , Hardy and Littlewood conjectured that when n tends to infinity, R(n) tends asymptotically to (i.e., the ratio of the two functions tends to one) n p-1 N2(n) = 2 C PRODUCT - , twin (log n)(log n-2) p odd prime p-2 divisor of n where p(p-2) C = PRODUCT - = twin p odd prime (p-1)^2 is the twin primes constant. In

49. Twin Prime Conjecture -- From MathWorld There are two related conjectures, each called the twin prime conjecture. As a result, the paper was retracted and the twin prime conjecture remains http://mathworld.wolfram.com/TwinPrimeConjecture.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 ... Unsolved Problems Twin Prime Conjecture 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 (Guy 1994, p. 19). It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true. While Hardy and Wright (1979, p. 5) note that "the evidence, when examined in detail, appears to justify the conjecture," and Shanks (1993, p. 219) states even more strongly, "the evidence is overwhelming," Hardy and Wright also note that the proof or disproof of conjectures of this type "is at present beyond the resources of mathematics." Arenstorf (2004) published a purported proof of the conjecture (Weisstein 2004). Unfortunately, a serious error was found in the proof. As a result, the paper was retracted and the twin prime conjecture remains fully open.

50. The Top Twenty: Twin Primes In 1919 Brun showed that the sum of the reciprocals of the twin primes converges to Two HardyLittlewood Conjectures; The Prime Glossary s twin primes http://primes.utm.edu/top20/page.php?id=1

51. Twin Prime Conjecture - Wikipedia, The Free Encyclopedia The twin prime conjecture is a famous problem in number theory that involvesprime numbers. There is also a generalization of the twin prime conjecture, http://en.wikipedia.org/wiki/Twin_Prime_Conjecture

Twin prime conjecture From Wikipedia, the free encyclopedia. (Redirected from Twin Prime Conjecture 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 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 Contents Partial results Hardy-Littlewood conjecture Serious problem found in potential proof See also ... edit 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 Paul Erdős showed that there is a constant c p such that p p c p , where p ' denotes the next prime after p . This result was successively improved; in

52. Goldbach's Conjecture - Wikipedia, The Free Encyclopedia The strong Goldbach conjecture is in fact very similar to the twin prime conjecture,and the two conjectures are believed to be of roughly comparable http://en.wikipedia.org/wiki/Goldbach's_conjecture

Goldbach's conjecture From Wikipedia, the free encyclopedia. In mathematics, Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics . It states: Every even integer greater than 2 can be written as the sum of two primes . (The same prime may be used twice.) For example, etc. Contents Origins Heuristic justification Rigorous results Trivia ... edit Origins In , the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture: Every integer greater than 2 can be written as the sum of three primes. He considered 1 to be a prime number , a convention subsequently abandoned. So today, Goldbach's original conjecture would be written: Every integer greater than 5 can be written as the sum of three primes. Euler, becoming interested in the problem, answered with an equivalent version of the conjecture: Every even number greater than 2 can be written as the sum of two primes. The former conjecture is today known as the "ternary" Goldbach conjecture, the latter as the "strong" Goldbach conjecture. The conjecture that all odd numbers greater than 9 are the sum of three odd primes is called the "weak" Goldbach conjecture. Both questions have remained unsolved ever since, although the weak form of the conjecture is much closer to resolution than the strong one. edit Heuristic justification The majority of mathematicians believe the conjecture (in both the weak and strong forms) to be true, at least for

53. Twin Prime Conjecture -- Facts, Info, And Encyclopedia Article twin prime conjecture. Categories conjectures, Number theory, The twinprime conjecture is a famous problem in (Click link for more info and facts http://www.absoluteastronomy.com/encyclopedia/t/tw/twin_prime_conjecture.htm

Twin prime conjecture [Categories: Conjectures, Number theory, Analytic number theory] The twin prime conjecture is a famous problem in (Click link for more info and facts about number theory) number theory that involves (An integer that has no integral factors but itself and 1) prime number s. 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 (Click link for more info and facts about twin prime) 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 1849 (Click link for more info and facts about de Polignac) 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 1915, (Click link for more info and facts about Viggo Brun) 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

54. Twin Prime -- Facts, Info, And Encyclopedia Article A strong form of the twin Prime conjecture, the (Click link for more info and The limiting value of f(x) is conjectured to equal the twin prime constant http://www.absoluteastronomy.com/encyclopedia/t/tw/twin_prime.htm

Twin prime [Categories: Number theory] A twin prime is a (An integer that has no integral factors but itself and 1) prime number that differs from another prime number by (The cardinal number that is the sum of one and one or a numeral representing this number) 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 (Click link for more info and facts about number theory) number theory for many years. This is the content of the (Click link for more info and facts about Twin Prime Conjecture) Twin Prime Conjecture . A strong form of the Twin Prime Conjecture, the (Click link for more info and facts about Hardy-Littlewood conjecture) Hardy-Littlewood conjecture , postulates a distribution law for twin primes akin to the (Click link for more info and facts about prime number theorem) prime number theorem Using his celebrated (Click link for more info and facts about sieve method) sieve method (Click link for more info and facts about Viggo Brun) Viggo Brun shows that the number of twin primes less than x is < x/(log x) . This result implies that the (A quantity of money) sum of the (Something (a term or expression or concept) that has a reciprocal relation to something else)

55. PlanetMath: Twin Prime Conjecture The case $ n=1$ is the twin prime conjecture. twin prime conjecture isowned by alozano. full author list (2) owner history (1) http://planetmath.org/encyclopedia/TwinPrimeConstant.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About twin prime conjecture (Conjecture) Two consecutive odd numbers which are both prime are called twin primes, e.g. 5 and 7, or 41 and 43, or 1,000,000,000,061 and 1,000,000,000,063. But is there an infinite number of twin primes ? In 1849 de Polignac made the more general conjecture that for every natural number , there are infinitely many prime pairs which have a distance of . The case is the twin prime conjecture. constant and infinitely many primes such that where denotes the next prime after . This result was improved in 1986 by Maier; he showed that a constant can be used. The constant is called the twin prime constant. In 1966, Chen Jingrun showed that there are infinitely many primes such that is either a prime or a semiprime "twin prime conjecture" is owned by alozano full author list owner history view preamble View style: HTML with images page images TeX source See Also: prime triples conjecture Also defines: twin prime constant Cross-references: semiprime constant distance natural number ... odd numbers There are 2 references to this object.

56. Read About Twin Prime Conjecture At WorldVillage Encyclopedia. Research Twin Pri twin prime conjecture. Everything you wanted to know about twin prime conjecturebut had no clue how to find it.. Learn about twin prime conjecture here! http://encyclopedia.worldvillage.com/s/b/Twin_Prime_Conjecture

Culture Geography History Life ... WorldVillage Twin prime conjecture From Wikipedia, the free encyclopedia. (Redirected from Twin Prime Conjecture 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 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 Contents 1 Partial results 2 Hardy-Littlewood conjecture 3 Serious problem found in potential proof 4 See also ... edit 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 p , where p ' denotes the next prime after p . This result was successively improved; in

57. Conjecture 3. Twin Prime's Conjecture This is the twin prime conjecture , which can be paraphrased this way Thereare infinite in the proof twin prime conjecture and so on, publish it. http://www.primepuzzles.net/conjectures/conj_003.htm

Conjectures Conjecture 3. Twin Prime's Conjecture If we define d n as : d n = p n+1 - p n , is easy to see that d =1 and d n Now, that " for n>1, dn=2 infinitely often" (Ref. 2, p. 19). This is the "Twin prime Conjecture", which can be paraphrased this way : "There are infinite consecutive primes differing by 2". SOLUTION Mr Liu Fengsui has sent (3/9/01) an argument that proves - according to him - the well known and named " k-tuple conjecture " This conjecture can be expressed the following way (see " Any admissible constellation of primes occurs infinitely often ". Therefore, if this the Mr Liu's argument is correct then also the Twin Primes conjecture has been proved. As you soon will discover this argument is close related to the Liu's approach to the prime numbers definition, approach that has been exposed in detail in the Problem 37 of these pages. What follows is Mr Liu's argument. I should strongly point out that the most that Mr. Liu

58. Twin Prime Conjecture Encyclopedia twin prime conjecture. The twin prime conjecture is a famousproblem in number theory that involves prime numbers. It states http://encyclopedie-en.snyke.com/articles/twin_prime_conjecture.html

Twin prime conjecture 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 1 Partial results 2 Hardy-Littlewood conjecture 3 Serious problem found in potential proof 4 See also ... 5 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

59. *Concerns Of Young Mathematicians* || Twin Prime Conjecture Proven? twin Prime conjecture Proven? Research. By kroth Posted Sat May 29th, twin Prime conjecture Proven? 4 comments (3 topical, 1 editorial, http://concerns.youngmath.net/story/2004/5/29/131042/934

The Young Mathematicians' Network Serving the Community of Young Mathematicians Sections: Front Page News Grad Life Undergrad Life ... All Twin Prime Conjecture Proven? Research By kroth Posted Sat May 29th, 2004 at 13:10:42 PDT There is an article on slashdot discussing a preprint of a paper by R. F. Arenstorf proving the twin primes conjecture. Read about it here The discussion that follows also has some amusing points, but not a whole lot of math. Of course you could also skip the article and just look at the preprint on arxiv here Update [2004-6-10 13:39:44 by overconvergent]: The preprint has been withdrawn from the arXiv; because: A serious error has been found in the paper, specifically, Lemma 8 is incorrect. Post a Comment Unions Considered Harmful? New Largest Prime Number Found (0 comments) Display: Threaded Minimal Nested Flat Sort: Unrated, then Highest Highest Rated First Lowest Rated First Ignore Ratings Newest First Oldest First Twin Prime Conjecture Proven? comments (3 topical, 1 editorial, hidden) [new] There seems to be some doubt emerging ...

60. Prime Strings, Goldbach And His Evil Twin Essentially, the twin prime conjecture stipulates that the binary string created The twin prime conjecture states that binary row in the first row is an 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:

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