81. Mathematics Resources Mathematicians close to cracking twin prime conjecture. Source Adfero Ltd.A group of international mathematicians has made major advances towards cracking http://www.scenta.co.uk/scenta/library/mathematics.cfm?cit_id=21556&FAArea1=cust

82. Isabel’s Math Blog » Prime Twin Conjecture Prime twin conjecture. Filed under. Number theory. — Izzy @ 1125 pm. The Primetwin conjecture has been (tentatively) proven by RF Arenstorf at Vanderbilt http://www.izzycat.org/math/index.php?p=23

83. MercuryNews.com 05/25/2005 Mathematician Lauded For (corrected Goldston s work involves the twin prime conjecture the idea that there are aninfinite While the new work doesn t prove the twin Prime conjecture, http://www.mercurynews.com/mld/mercurynews/news/11732485.htm

84. NMBRTHRY Archives - June 2004 Arenstorf s paper on the twin Prime conjecture (14 lines) From PieterMoree moree@science.uva.nl ; Re Arenstorf s paper on the twin Prime conjecture (22 http://listserv.nodak.edu/cgi-bin/wa.exe?A1=ind0406&L=nmbrthry&D=0

85. NMBRTHRY Archives -- August 2002 (#18) in agreement with the twin prime conjecture prediction and my estimation of Brun sconstantis. B = 1.90216 05831 04 The results are in perfect agreement http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0208&L=nmbrthry&P=1968

86. Welcome To Mathsoft It s intriguing that both the Extended twin Prime conjecture and the ExtendedGoldbach conjecture involve the same constant, Ctwin. http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/Number_Theory_Cons

Company Press Room Partners Events ... Math Resources Print this page Hardy-Littlewood Constants The sequence of prime numbers p = 2, p = 3, p = 5, p = 7, p = 11, p = 13, p Extended Twin Prime Conjecture where Conjectures involving two different kinds of prime triples Conjectures involving two different kinds of prime quadruples where Extended Goldbach Conjecture if R(n) is defined to be the number of representations of an even integer n as the sum of two primes (order counts), then It's intriguing that both the Extended Twin Prime Conjecture and the Extended Goldbach Conjecture involve the same constant, C twin The Hardy-Littlewood constants discussed above all involve infinite products over primes. Other such products occur in our essays on Artin's constant Landau-Ramanujan constant Pell-Stevenhagen constants Hafner-Sarnak-McCurley constant Euler Totient Function Asymptotic constants Infinite series over primes are the main topic in Meissel-Mertens constants and in Brun's constant . We will mention C twin again in connection with Artin's constant since the two constants are quite similar. Linnik's constant also involves prime numbers.

87. Math Trek: Prime Twins, Science News Online, June 2, 2001 Indeed, the twin prime conjecture is considered one of the major unsolved problemsin number theory. It was even mentioned in the 1996 movie A Mirror Has http://www.sciencenews.org/articles/20010602/mathtrek.asp

Science News Books. Subscribe to Science News ... Science News for Kids Math Trek Prime Twins Food for Thought Dietary protection against sunburn (with recipe) Science Safari Daily Planet Earth TimeLine 70 Years Ago in Science News Science News e-LETTER. ... Week of June 2, 2001; Vol. 159, No. 22 Prime Twins Ivars Peterson Number theory offers a host of problems that are remarkably easy to state but fiendishly difficult to solve. Many of these questions and conjectures feature prime numbers—integers evenly divisible only by themselves and 1. For instance, primes often occur as pairs of consecutive odd integers: 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. So-called twin primes are scattered throughout the list of all prime numbers. There are 16 twin prime pairs among the first 50 primes. The largest known twin prime is the 32,220-digit pair 318032361 x 2 +/–1, found recently by David Underbakke and Phil Carmody. Although most mathematicians believe that there are infinitely many twin primes, no one has yet proved this conjecture to be true. Indeed, the twin prime conjecture is considered one of the major unsolved problems in number theory. It was even mentioned in the 1996 movie A Mirror Has Two Faces , which starred Barbra Streisand.

88. Slashdot | Twin Prime Proof Proffered twin Prime Proof Proffered article related to Education and Science. there was a proposed proof to the Poincare conjecture not the Perelman proof http://science.slashdot.org/article.pl?sid=04/11/04/0544229&tid=146&tid=14

89. Slashdot | There Are Infinitely Many Prime Twins twin primes are pairs of primes where both p and p + 2 are prime. It is conjecturedthat there are an infinite number of twin primes but proving this http://science.slashdot.org/article.pl?sid=04/05/28/2012209&tid=146

90. Page 015 and the twin prime conjecture are almost true, in the following sense. 1.There exist an integer N such that every even integer greater than N is the http://www.math.utoledo.edu/~jevard/Page015.htm

Almost twin primes and Chen's theorem Page maintained by Jean-Claude Evard. Last update: June 23, 2003. AMS classification numbers 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. Definitions 1. We say that an integer greater than 1 is an r-almost prime if and only if it is the product of at most r primes. 2. The set of all r-almost primes is denoted by P r 3. We say that a positive integer is an almost prime if and only if it is a 2-almost prime 4. We say that a pair of positive integers are r-almost twin primes if and only if one integer of the pair is a prime and the other is an r-almost prime. 5. We say that a pair of positive integers are almost twin primes if and only if they are 2-almost twin primes Historical result about almost twin primes Chen's theorem says that both the Goldbach's conjecture and the twin prime conjecture are almost true, in the following sense:

91. American Institute Of Mathematics Number theorists moved a step closer to the resolving the twin prime conjecturethis week when a new paper appeared on the internet, see the AIM preprint, http://aimath.org/primegaps/

Information: Background News Governance Sponsors and ... Staff ARCC: Upcoming Workshops Outlines of open problems Participant resources AIM Activities: Preprints submitting Conferences Projects ... Visitors May 2005: Breakthrough in Prime Number theory Number theorists moved a step closer to the resolving the twin prime conjecture this week when a new paper appeared on the internet, see the AIM preprint , which gives a proof that the spacing between consecutive primes is sometimes very much smaller than the average spacing. This result was originally reported two years ago by Dan Goldston and Cem Yildirim (see the original AIM press release and technical description ) but was later retracted . Now, Janos Pintz has joined the team and completed the proof of this important result. Amazingly, the new proof can be given with full details in about 8 pages. Moreover, the techniques used are familiar to number theorists. The earlier version involved some new methods which turned out to be incorrect. There is a belief among some number theorists that a psychological barrier has been broken and that a proof of the twin prime conjecture may not be far away. Indeed, Goldston expressed such a belief during a presentation of this new work at AIM on May 24. The precise statement of the new theorem is that for any positive number there exist primes p and p' such that the difference between p and p' is smaller than . The proof of an even stronger statement, namely that the difference can be as small as

92. [math/0405509] There Are Infinitely Many Prime Twins There Are Infinitely Many Prime Twins. Authors RF Arenstorf Comments This paperhas been withdrawn Subjclass Number Theory MSC-class 11A41; 11N05 http://arxiv.org/abs/math.NT/0405509/

Mathematics, abstract math.NT/0405509 From: Richard Arenstorf [ view email ] Date ( ): Wed, 26 May 2004 20:22:18 GMT (28kb) Date (revised v2): Tue, 8 Jun 2004 22:51:33 GMT (0kb,I) There Are Infinitely Many Prime Twins Authors: R. F. Arenstorf Comments: This paper has been withdrawn Subj-class: Number Theory MSC-class: A serious error has been found in the paper, specifically, Lemma 8 is incorrect. Full-text: Source References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math find abs

93. Foreign Dispatches: An Elementary Proof Of The Prime Number Theorem http://foreigndispatches.typepad.com/dispatches/2005/06/an_elementary_p.html

Foreign Dispatches Random Remarks on Current Affairs Advertising Periodicals Asahi (English) Daily Star (Lebanon) Der Spiegel Die Zeit ... Yomiuri (English) Math/IT ACM AMS Acta Arithmetica Annals of Mathematics ... PARC - User Interfaces References Classic Works of Japanese Literature Government Accountability Office International Monetary Fund MIT OpenCourseWare ... World Bank Politics American Enterprise Institute Brookings Institution Cato Institute Centre for the New Europe Blogs A Different Opinion A Fistful of Euros Adam Smith Institute AdamSmithee ... Winds of Change Sitemeter var site="s12fgndisp" Math Books K. F. Ireland and M. Rosen: A Classical Introduction to Modern Number Theory Herbert B. Enderton: A Mathematical Introduction to Logic Robin Hartshorne: Algebraic Geometry G. H. Hardy, E.M. Wright: An Introduction to the Theory of Numbers ... Main June 15, 2005 An Elementary Proof of the Prime Number Theorem I was wondering if I could find an online copy of Paul Erd¶s' 1949 paper containing a proof, using only "elementary" methods (and itself leaning on Atle Selberg 's elementary proof of a certain asymptotic formula), of the

94. The Mathematical Tourist These twins are spread throughout the list of known prime numbers. To try toprove many conjectures about prime numbers, mathematicians therefore often http://www.fortunecity.com/emachines/e11/86/tourist2b.html

web hosting domain names photo sharing Prime Pursuits Ivars Peterson PRIME PROPERTIES The study of prime numbers has long been a central part of number theory, a field traditionally pursued for its own sake and for the beauty of its results . Once thought to be the purest of pure mathematics, this ancient pastime now figures prominently in modern computer science. The security of modern cryptosystems depends very strongly on the twin questions of how easy it is to identify primes and how hard it is to factor a large, random number. Neither question has a clear answer yet. Divisible evenly only by themselves and the number 1, the primes stand at the center of number theory. Like chemical elements in chemistry or fundamental particles in physics, they are building blocks in the mathematics of whole numbers. All other whole numbers, known as composites, can be written as the product of smaller prime numbers. In fact, according to the fundamental theorem of arithmetic , each composite number has a unique set of prime factors. Hence, the composite number 20 can be broken down into the prime factors 2, 2, and 5. No other composite number has the same set of factors. The number 1 is considered to be neither prime nor composite.

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