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# Topic: Twin prime conjecture

###### In the News (Mon 20 May 13)

 Conjecture In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has been able to prove or disprove. When a conjecture has been proven to be true, it becomes known as a theorem, and joins the realm of mathematical facts. Although many of the most famous conjectures have been tested across an astounding range of numbers, this is no guarantee against a single counterexample, which would immediately disprove the conjecture. www.ebroadcast.com.au /lookup/encyclopedia/co/Conjecture.html   (448 words)

 Twin prime It is unknown whether there exist infinitely many twin primes, but most number theorists believe this to be true. 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. It is known that the sum of the reciprocals of all twin primes converges (see Brun's constant). www.ebroadcast.com.au /lookup/encyclopedia/tw/Twin_prime.html   (196 words)

 Prime Strings, Goldbach and his Evil Twin The new string in shift 2 should now show all of the twin primes, the rows in the row labeled shift 4 shows all of the primes that differ by four, the string in the row labeled shift 6 shows the primes that are 6 apart, etc.. The twin prime conjecture states that binary row in the first row is an infinite decimal. The n-prime conjecture would extends the twin prime conjecture—speculating that all of the rows are infinite decimals. descmath.com /prime/prime_strings.html   (2222 words)

 Prime Producing Equations Prime and composite numbers exhibit the most organization in the classical odd number pyramid, and in the number field generated by extending the rows of the pyramid to the right and left of the pyramid’s boundaries. The diagonals of primes, in fl (from top to bottom), are outputs from the quadratic equation f(x) = x^2 - x + c where c = 11, c = 17, and c = 41. All the primes except 2, 3, 5, and 7 are present at c = 11 + d*30 or c = 17 + d*30 (every 30th column from c = 11 and 17). www.prime-equations.com   (784 words)

 Conjecture 3. Twin Prime's Conjecture This criterion R(k,a) recursively enumerate all prime k-tuples. As each further prime is cast out, a proportion of the u integers become m integers and the number of possible twin primes decreases. twin primes between a prime and its square, and since there are an infinite number of primes, there must be an infinite number of twin primes. www.primepuzzles.net /conjectures/conj_003.htm   (3798 words)

 Wikinfo | Conjecture   (Site not responding. Last check: 2007-10-03) In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. Once a conjecture has been proven, it becomes known as a theorem, and it joins the realm of mathematical facts. The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be undecidable (or independent) from the generally accepted set of axioms of set theory. www.wikinfo.org /wiki.php?title=Conjecture   (723 words)

 Twin prime conjecture - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-03) Defining a Chen prime to be a prime p such that p + 2 is either a prime or a semiprime, Terence Tao and Ben Green showed in 2005 that there are infinitely many three term arithmetic progressions of Chen primes. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. en.wikipedia.org.cob-web.org:8888 /wiki/Twin_prime_conjecture   (572 words)

 Introduction to twin primes and Brun's constant computation 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 [5]). Based on heuristic considerations, a law (the twin prime conjecture) was developed, in 1922, by Godfrey Harold Hardy (1877-1947) and John Edensor Littlewood (1885-1977) to estimate the density of twin primes. According to this conjecture the density of twin primes is equivalent to the density of cousin primes. numbers.computation.free.fr /Constants/Primes/twin.html   (1986 words)

 Prime number In mathematics, a prime number, or prime for short, is a natural number larger than 1 that has as its only positive divisors (factors) 1 and itself. Twin Prime Conjecture: A twin prime is a pair of primes with difference 2, such as 11 and 13. Prime ideals are an important tool and object of study in commutative algebra and algebraic geometry. www.fastload.org /pr/Prime_number.html   (1738 words)

 The Top Twenty: Twin Primes Twin primes are pairs of primes which differ by two. It has been conjectured (but never proven) that there are infinitely many twin primes. Here the infinite product is the twin prime constant (estimated by Wrench and others to be approximately 0.6601618158...), and we introduce an integral to improve the quality of the estimate. primes.utm.edu /top20/page.php?id=1   (342 words)

 Prime Finding: Mathematicians mind the gap: Science News Online, March 29, 2003   (Site not responding. Last check: 2007-10-03) 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. 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. The distribution of primes is closely related to one of the most renowned questions in mathematics, the Riemann hypothesis, which concerns an infinite sum called the zeta function. www.sciencenews.org /20030329/fob1.asp   (669 words)

 PBS Discussions :: View topic - Twin Prime Conjecture Primes become less frequent among progressively larger numbers, and the average distance between prime numbers appears to increase in proportion to something like the square root of the location in the number sequence. Recognizing that all primes after 3 must be of the form 6n-1 or 6n+1, one sees that all that the sieving process does is kill off either one or both of the numbers defined by this formula. As to Euclid’s relationship to Twin Primes, there is debate over whether he recognized them first, but it is fair to say, as Dr. Goldston has pointed out, that he did not explicitly formulate the conjecture in its modern form. discussions.pbs.org /viewtopic.pbs?t=45116   (1801 words)

 San José State News Twin primes are two prime numbers that differ by 2, such as 11 and 13 or 29 and 31. The twin-prime conjecture is that there are an infinite number of twin primes. Goldston has been studying twin primes and more generally the gaps between primes since he was a graduate student at Berkeley in the late 1970s. www.sjsu.edu /news/news_detail.jsp?id=1640   (356 words)

 PlanetMath: Sophie Germain prime It is conjectured that there are infinitely many Sophie Germain primes, but (like the Twin Prime Conjecture) this has not been proven. Cross-references: twin prime constant, twin prime conjecture, prime number This is version 5 of Sophie Germain prime, born on 2004-09-03, modified 2006-09-01. planetmath.org /encyclopedia/GermainPrime.html   (72 words)

 twin primes Pairs of prime numbers that differ by two, the first of which are 3 and 5, 5 and 7, 11 and 13, and 17 and 19. In 1919 Brun showed that the sum of the reciprocals of the twin primes converges to a sum now known as Brun's constant: (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) +... The twin-prime conjecture generalizes to prime pairs that differ by any even number n, and generalizes even further to certain finite patterns of numbers separated by specified even differences. www.daviddarling.info /encyclopedia/T/twin_primes.html   (334 words)

 BBC NEWS | Science/Nature | Prime number breakthrough Primes can be multiplied to obtain all of the other integers. A curious observation is that primes occur in twins with a surprising regularity. The distribution of primes is closely related to one of the most renowned unsolved questions in mathematics, the Riemann hypothesis, which concerns an infinite sum of numbers called the zeta function. news.bbc.co.uk /1/hi/sci/tech/2911945.stm   (591 words)

 Twin prime - Wikipedia, the free encyclopedia A twin prime is a prime number that differs from another prime number by two. Some examples of twin prime pairs are 5 and 7, 11 and 13, and 821 and 823. Every twin prime pair greater than 3 is of the form (6n − 1, 6n + 1) for some natural number n, and with the exception of n = 1, n must end in 0, 2, 3, 5, 7, or 8. en.wikipedia.org /wiki/Twin_prime   (505 words)

 Science Journal: Prime-number proofs chalk up more success   (Site not responding. Last check: 2007-10-03) But now that primes are the basis for codes that encrypt financial data as well as national-security transmissions, making sure that mathematicians' hunches about primes are actually true matters in the real world, too. A prime is a number whose only factors are 1 and itself, like 2, 3, 5, 7 and 18,793, not to mention 2 multiplied by itself 30,402,457 times minus 1 (the largest prime discovered so far, announced on Christmas; testing ever larger numbers for primeness, while doable, is very time consuming). Their proof suggests that there are infinitely many consecutive primes that differ by only 16, which is getting close to the 2 claimed by the twin-prime conjecture. www.post-gazette.com /pg/06034/649668.stm   (854 words)

 PlanetMath: twin prime conjecture Two consecutive odd numbers which are both prime are called twin primes, e.g. But is there an infinite number of twin primes ? This is version 8 of twin prime conjecture, born on 2003-01-07, modified 2006-10-04. planetmath.org /encyclopedia/TwinPrimeConstant.html   (137 words)

 Several Proofs of the Twin Primes Conjecture are not all the twin prime intervals, i.e., Because all numbers n + k and n - k are not prime numbers, equations (2)-(4) are necessary for proving the infinitude but are not sufficient for finding the existence of prime numbers at these locations. However, since the number of twin primes is unlimited, their sum is an even number 2n unlimited and every sum of primes is an even number. www.coolissues.com /mathematics/Tprimes/tprimes.htm   (508 words)

 The Year in Science: Mathematics - news education science magazines technology science news environment magazine ...   (Site not responding. Last check: 2007-10-03) The new result is closely related to the famous twin prime conjecture, which says there are an infinite number of pairs of prime numbers that differ only by two. Primes are numbers that can be divided only by themselves or by 1 without leaving a remainder. The smallest twin primes are 3 and 5; 11 and 13 provide another example. www.discover.com /issues/jan-06/features/mathematics   (598 words)

 American Institute of Mathematics   (Site not responding. Last check: 2007-10-03) 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. 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. This work is a major step toward the centuries-old problem of showing that there are infinitely many "twin primes": prime numbers which differ by 2, such as 11 and 13, 17 and 19, 29 and 31,... aimath.org /primegaps   (454 words)

 Twin Primes Conjecture / Prime Sieve Notes: At this point, all primes of this mathematical form are potentially twin, as each is spaced 2 apart from its nearest neighbor. There is one more thing that makes the twin prime sieve different from the regular prime sieve. The Twin Prime procedure takes these SAME branches, and pares off the ones that do not contain twin primes. members.aol.com /scirealm/TwinPrimes.html   (823 words)

 The Prime Glossary: twin prime conjecture   (Site not responding. Last check: 2007-10-03) The (weak) twin prime conjecture is that there are infinitely many twin primes. twin primes less than or equal to x. The constant written above as an infinite product is the twin primes constant. primes.utm.edu /glossary/page.php?sort=TwinPrimeConjecture   (104 words)

 Twin Prime Conjecture -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-03) There are two related conjectures, each called the twin prime conjecture. As a result, the paper was retracted and the twin prime conjecture remains fully open. An extended form of this conjecture, sometimes called the strong twin prime conjecture (Shanks 1993, p. mathworld.wolfram.com.cob-web.org:8888 /TwinPrimeConjecture.html   (406 words)

 Ivars Peterson's MathTrek -Prime Twins Indeed, the twin prime conjecture is considered one of the major unsolved problems in number theory. Kelly and Pilling liken the numerical behavior of twin primes to that a radioactive substance, where the likelihood of one of its atoms decaying in any short time interval is fixed. For example, the distance between the twins (29, 31) and (17, 19) is 31 – 19, or 12. www.maa.org /mathland/mathtrek_6_4_01.html   (950 words)

 The music of primes   (Site not responding. Last check: 2007-10-03) The primes are the key to their survival [link to prime number cicadas] http://www.cubethemovie.com The Conjecture says that you will always find clusters of primes where N and N+2 are both prime. The first twin that Streisand would find beyond a million is the pair of primes 1,000,037 and 1,000,039. www.musicoftheprimes.com /films.htm   (530 words)

 Page 012 The twin prime conjecture and extensions of Chen's theorem from the d-twin prime conjecture of Hardy and Littlewood. Lavrik, On the twin prime hypothesis of the theory of primes www.math.utoledo.edu /~jevard/Page012.htm   (1360 words)

 Number Theory The basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs (p,p+2) in which both p and p+2 are prime integers. An interesting consequence of this conjecture is the dependence of the solution on abundant numbers; an abundant number is an integer whose sum of its proper divisors exceeds the integer. www.math.utah.edu /~gold/numbertheory.html   (872 words)

 Platonism, Intuition and the Nature of Mathematics. By K.Podnieks Discussing the amount of prime numbers they believed that they are discussing objects as real as collections of things in their everyday practice. twin prime conjecture), yet the problem remains unsolved up to day. And you meet twin pairs in it from time to time: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43),... linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt1.html   (4118 words)

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