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Topic: Hardy-Littlewood conjecture

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Search Results for conjecture* This conjecture became known as "the main conjecture on cyclotomic fields" and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves. In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all g(x) which have a maximum value and decrease monotonically on either side of this maximum. Burnside conjectured that every finite group of odd order is soluble and it is not surprising that he failed to prove this result as it was not proved until 1962 when W Feit and J C Thompson proved the result in a 300 page paper. www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=conjecture*&CONTEXT=1   (9425 words)

Admissible prime constellations In the same memoir [1], Hardy and Littlewood conjectured, based on empirical evidence, that rho(x) <= pi(x) for x >= 2, where rho(x) is the largest number of primes that occur indefinitely often in an interval of length x. This second conjecture is connected with another conjecture of Hardy and Littlewood, which states that pi(x+y) - pi(x) <= pi(y) for x,y >= 2, i.e., it states that no interval with n>1 consecutive integers can contain more primes than the interval [1,n]. This conjecture is known as the prime k-tuple conjecture. www.ieeta.pt /~tos/apc.html   (1090 words)

Riemann hypothesis - LearnThis.Info Enclyclopedia Later work by Hardy and Littlewood in 1921 and by Selberg in 1942 gave estimates for the average density of zeros on the critical line. Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he did not attempt a proof. The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeross of Riemann's zeta function ζ(s);. encyclopedia.learnthis.info /r/ri/riemann_hypothesis.html   (1090 words)

Twin prime conjecture Hardy and John Littlewood), which is concerned with the distribution of twin primes, in analogy to the prime number theorem. The twin prime conjecture is a famous problem in number theory that involves prime numbers. In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs which have a distance of 2k. www.worldhistory.com /wiki/T/Twin-prime-conjecture.htm   (1090 words)

Read Poincaré couldn't have made this conjecture absent his years of study of topology, or the earlier theorems he'd carefully proved, or the earlier conjectures on the same theme he'd tried out and found to be false. And it's the Poincaré conjecture that, courtesy of the Clay Foundation, carries a million-dollar bounty. The British number theorist G.H. Hardy, in A Mathematician's Apology, one of the most widely read books about the nature and practice of mathematics, famously wrote: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." slate.msn.com /toolbar.aspx?action=read&id=2082960   (1268 words)

A Generalization of the Prime Pairs / Hardy-Littlewood / Dickson / Schinzel-Sierpinski Conjectures In fact, the Hardy-Littlewood Conjecture holds that the density of such prime k-tuples is asymptotic to... Dickson's Conjecture, which is discussed in more detail here in Chris Caldwell's informative nest of pages on prime numbers, extends the conjecture to a k-tuple of linear equations with integer coefficients, i.e., a In that case, the suitable reduction condition permits N to be "suitably reduced" by division by 2, and N+1 to be "suitably reduced" by division by 3. www.d.umn.edu /~schilton/Articles/primep~1.htm   (1465 words)

sci.math Message Message: Re: Hardy & Littlewood's Partitio Numerorum III, Conjecture H Subject: Re: Hardy & Littlewood's Partitio Numerorum III, Conjecture H > > > In Hardy and Littlewood's Partitio Numerorum III, Conjecture H is that every > > sufficiently large integer is either a square or the sum of a square and a > > prime. mathforum.com /discuss/sci.math/m/420073/420075   (1465 words)

Hardy-Littlewood conjecture Please See Twin prime conjecture For Further Information about Hardy-Littlewood conjecture. www.bambooweb.com /articles/h/a/Hardy-Littlewood_conjecture.html   (1465 words)

Mathsoft: Number Theory Constants: Hardy-Littlewood Constants We focus on certain heuristic formulas, developed by Hardy and Littlewood. Conjecture involving primes of the form m² + 1 Conjectures involving two different kinds of prime triples www.mathsoft.com /mathresources/constants/numbertheory/article/0,,2000,00.html   (1465 words)

John Edensor Littlewood Littlewood's inequality on bilinear forms was a forerunner of the later Grothendieck tensor norm theory. Littlewood was born in Rochester in Kent, and studied at Cambridge University. In his other work Littlewood collaborated with Paley in Fourier theory, and with Offord in combinatorial work on random sums, in developments that opened up fields still intensively studied. www.worldhistory.com /wiki/J/John-Edensor-Littlewood.htm   (1465 words)

Second Hardy-Littlewood conjecture In number theory, the second Hardy-Littlewood conjecture concerns the number of primess in intervals. This is probably false in general as it is inconsistent with the first Hardy-Littlewood conjecture, but the first violation is likely to occur for very large values of x and y. If π( x) is the number of primes up to and including x then the conjecture states that www.sciencedaily.com /encyclopedia/second_hardy_littlewood_conjecture   (1465 words)

Timeline of mathematics 2004 - Richard Arenstorf provides proofs of twin prime conjecture and Hardy-Littlewood conjecture which contain an error in Lemma 8, discovered by Michel Balazard, 1999 - the full Taniyama-Shimura conjecture is proved. 1796 - Adrien-Marie Legendre conjectures the prime number theorem, www.worldhistory.com /wiki/T/Timeline-of-mathematics.htm   (1465 words)

Riemann hypothesis - Wikipedia, the free encyclopedia The traditional formulation of the Riemann hypothesis obscures somewhat the true importance of the conjecture. On the other hand, De Branges's successful proof of the Bieberbach conjecture was also preceded by his failed proofs of it. In June 2004, Louis De Branges de Bourcia of Purdue University, the same mathematician who solved the Bieberbach conjecture, claimed to have proved the Riemann hypothesis in an "Apology for the proof of the Riemann Hypothesis" en.wikipedia.org /wiki/Riemann_hypothesis   (1465 words)

Twin prime conjecture - Wikipedia, the free encyclopedia The numerical evidence behind the Hardy-Littlewood conjecture is quite impressive. sieve theory, and he managed to treat the twin prime conjecture and This conjecture can be justified (but not proven) by assuming that 1 / ln  t describes the density function of the prime distribution, an assumption suggested by the prime number theorem. en.wikipedia.org /wiki/Twin_Prime_Conjecture   (1465 words)

Prime k-tuplets The most important unsolved conjecture of prime number theory, indeed, all of mathematics - the Riemann Hypothesis - asserts that the error term can be bounded by the function A sqrt( x) log x. The Partitio Numerorum : III paper [HL22] goes on to formulate a general conjecture concerning the distribution of arbitrary groups of prime numbers (The k -tuplets of this site are special cases): Let b 1, b 2,..., b k be k distinct integers. The nearest to proving the conjecture is Jing-Run Chen's result that there are infinitely many primes p such that p +2 is either prime or the product of two primes [HR73]. www.ltkz.demon.co.uk /ktuplets.htm   (1465 words)

A Generalization of the Prime Pairs / Hardy-Littlewood / Dickson / Schinzel-Sierpinski Conjectures [Spring 1999: I AM INDEBTED TO PROFESSOR WILLIAM ADAMS OF THE DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MARYLAND, FOR THE OBSERVATION THAT THE "GENERALIZATIONS" I PROPOSE DO NOT GO BEYOND DICKSON'S CONJECTURE, EVEN THOUGH THEY APPEAR TO. The general conjecture is that the same is true of any k-tuple with any pattern of suitable reductions. While there is but slight merit in thus plausibly extending these previous conjectures, I do so with the thought that the creation of as general a (true) conjecture as possible will direct us to an appropriate method of solution. www.d.umn.edu /~schilton/Articles/primep~1.htm   (1465 words)

E. H. LIEB -- Publications (with R. Seiringer) Equivalent forms of the Bessis-Moussa-Villani Conjecture, J. Stat. Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture, Adv. (with I. Affleck) A Proof of Part of Haldane's Conjecture on Spin Chains, Lett. www.math.princeton.edu /~lieb/publications.html   (1465 words)

Enumeration to 1.6*10^15 of the prime quadruplets In support of the accuracy and probable validity of the Hardy-Littlewood conjecture (2.3) for the quadruplets, note that delta_4(x) has about half as many digits as L_4(x). On the other hand, the particular scaling factor sqrt(x)*(ln(x))^2, taken from the conjectured error term in (2.12), is not indispensable to this line of error analysis. The validity of this conjecture will be central to our methods for estimating B_4 and its error bound. www.trnicely.net /quads/quads.html   (1465 words)

Admissible prime constellations This second conjecture is connected with another conjecture of Hardy and Littlewood, which states that pi(x+y) - pi(x) <= pi(y) for x,y >= 2, i.e., it states that no interval with n>1 consecutive integers can contain more primes than the interval [1,n]. In order to actually find a counter-example of the pi(x) conjecture, the first step is to find an interesting constellation. Assuming the truth of the prime k -tuple conjecture, rho(x)=s(x). www.ieeta.pt /~tos/apc.html   (1465 words)

Mathcad Library: Constants assuming the truth of a Hardy-Littlewood conjecture and based on a large dataset of twin primes. By "Hardy-Littlewood conjecture", we mean a certain formula that asymptotically describes the number of twin primes www.mathcad.com /library/constants/brun.htm   (1465 words)

Timeline of mathematics First mathematician to work on the details of 'Arithmetic and Algebra of inheritance' besides the systematisation of the theory of linear and quadratic equations. A Modern History of Blacks in Mathematics - A contemporary history of Blacks in Mathematics,featuring the first African Americans in the Mathematical Sciences and related events in the past 300 years. 2450 BC - Egypt, first systematic method for the approximative calculation of the circle on the basis of the Sacred triangle 3-4-5, www.nebulasearch.com /encyclopedia/article/Timeline_of_mathematics.html   (1465 words)

Isabel’s math blog » Number theory The Prime Twin Conjecture is the statement that there are infinitely many pairs of primes which differ by 2, such as (3,5), (5,7), (11,13), (17,19), (29,31), and so on. In fact, it wouldn’t surprise me if a lot of the really hard-to-prove but easily-stated results of number theory (for example, the Goldbach conjecture) are connected to the existing, proven body of mathematics by some very deep connection that we haven’t seen yet. Hardy once said (in his Mathematician’s Apology) that calculating the decimal digits of numbers constituted non-serious mathematics. www.izzycat.org /math/index.php?cat=4   (1465 words)

eisBTfry00096.txt %H A076439 E. Weisstein, The World of Mathematics: Pillai's Conjecture %H A076439 T. Noe, Unique solutions to Pillai's Equation requiring only squares for n %e A035789 The first lonely twin primes (A069453) are 29,31 (23 and 37 are non-twin), 41,43 (37 and 47 are non-twin), 59,61 (53 and 67 are non-twin). First complex prime is 1+i with 2 as corresponding real prime, according to reference, page 1-2. www.research.att.com /~njas/sequences/eisBTfry00096.txt   (1465 words)

Second Hardy-Littlewood conjecture - Wikipedia, the free encyclopedia This is probably false in general as it is inconsistent with the first Hardy-Littlewood conjecture, but the first violation is likely to occur for very large values of x and y. In number theory, the second Hardy-Littlewood conjecture concerns the number of primes in intervals. If π(x) is the number of primes up to and including x then the conjecture states that en.wikipedia.org /wiki/Second_Hardy-Littlewood_conjecture   (1465 words)

Earliest Known Uses of Some of the Words of Mathematics (C) The history of the contributing inequalities is given in Inequalities by G. Hardy, J. Littlewood and G. Polya (1934): the inequality for sums is due to A. Cauchy in 1821 and the inequality for integrals to H. CONJECTURE in the sense of "an opinion or supposition based on evidence which is admittedly insufficient" had been in English for a more than a century when Isaac Newton used the term in 1672: "I shall refer him to my former Letter, by which that conjecture will appear to be ungrounded." Mr. The earliest use of the noun conjecture in mathematical writing that I have encountered is in Hilbert's 1900 address, where it is used exactly once, in reference to Kronecker's Jugendtraum. members.aol.com /jeff570/c.html   (16673 words)

MainiAbstract.html A very general (and still almost completely open) conjecture concerning such structures is the prime tuple conjecture of Hardy and Littlewood. Among other things (for instance, the twin prime conjecture), it would imply that there are infinitely many arithmetic progressions in the primes of any specified length. The second (which is new), is that Szemeredi's theorem can be generalized to the statement that any subset of a "sufficiently pseudorandom set" with positive density contains arbitrarily long arithmetic progressions. www.maths.ed.ac.uk /~depiro/MainiAbstract.html   (207 words)

David O'Doherty Homepage The distribution of prime numbers, and an investigation of the second Hardy-Littlewood conjecture. I'm a second year undergraduate reading mathematics at Jesus College, The University of Cambridge. This is the project for which I won the Esat Young Scientist Exhibition. www.srcf.ucam.org /~dmo25   (179 words)

The Prime Glossary: arithmetic sequence Dickson's conjecture says the answer should be arbitrarly long--but finding long sequences of primes is quite difficult. It is conjectured that it actually occurs before k!+1 [Kra2005]. Littlewood, "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes," Acta Math. primes.utm.edu /glossary/page.php?sort=ArithmeticSequence   (516 words)

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