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# Topic: Conjecture

 The abc conjecture Assuming the Birch and Swinnerton-Dyer conjecture, it is shown in [Go-Sz] that this conjecture is equivalent to the Szpiro conjecture for modular elliptic curves. [Wa2] Walsh, P.G. On a conjecture of Schinzel and Tijdeman. The Wieferich criterion, the ABC conjecture and Shimura's correspondence, Satya Mohit, M.Sc. www.math.unicaen.fr /~nitaj/abc.html   (4294 words)

 Conjecture - Wikipedia, the free encyclopedia Once a conjecture is formally proven true it is elevated to the status of theorem and may be used afterwards without risk in the construction of other formal mathematical proofs. In scientific philosophy, Karl Popper pioneered the use of conjecture to indicate a statement which is presumed to be real, true, or genuine, mostly based on inconclusive grounds, in contrast with a hypothesis (hence theory, axiom, principle), which is a testable statement based on accepted grounds. Until recently, the most famous conjecture was the mis-named Fermat's last theorem, mis-named because although Fermat claimed to have found a clever proof of it, none could be found among his notes after his death. en.wikipedia.org /wiki/Conjecture   (758 words)

 Goldbach's conjecture - Wikipedia, the free encyclopedia In mathematics, Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. The former conjecture is today known as the "ternary" Goldbach conjecture, the latter as the "strong" or "binary" Goldbach conjecture. 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 difficulty. en.wikipedia.org /wiki/Goldbach's_conjecture   (1615 words)

 PlanetMath: ABC conjecture This conjecture was formulated by Masser and Oesterlé in 1980. The ABC conjecture is considered one of the most important unsolved problems in number theory, as many results would follow directly from this conjecture. This is version 13 of ABC conjecture, born on 2001-10-15, modified 2004-04-30. planetmath.org /encyclopedia/ABCConjecture.html   (125 words)

 The Prime Glossary: Goldbach's conjecture Goldbach wrote a letter to Euler dated June 7, 1742 suggesting (roughly) that every even integer is the sum of two integers p and q where each of p and q are either one or odd primes. Goldbach's conjecture: Every even integer n greater than two is the sum of two primes. Among other things, they conjectured that the number of ways of writing n as the sum of two primes, G(n), is asymptotic to twice the twin prime constant times n/(log n) primes.utm.edu /glossary/page.php?sort=GoldbachConjecture   (681 words)

 Goldbach conjecture In its original form, now known as the weak Goldbach conjecture, it was put forward by the Prussian amateur mathematician and historian Christian Goldbach (1690-1764) in a letter dated Jun. 7, 1742, to Leonhard Euler. In this guise it says that every whole number greater than 5 is the sum of three prime numbers. Euler restated this, in an equivalent form, as what is now called the strong Goldbach conjecture or, simply, the Goldbach conjecture: every even number greater than 2 is the sum of two primes. www.daviddarling.info /encyclopedia/G/Goldbach_conjecture.html   (375 words)

 index There is no known evidence of prior knowledge of Beal's conjecture and all references to it begin after Beal's 1993 discovery and subsequent dissemination of it. While Beal's conjecture was widely received with enthusiasm by the mathematics community at large, it seems that there are always people with other motivations. The Beal Conjecture is sometimes referred to as "Beal's conjecture", "Beal's problem" or the "Beal problem". www.bealconjecture.com   (857 words)

 Prime Conjectures and Open Question Ramaré95] (Goldbach's conjecture suggests two) and in 1966 Chen proved every sufficiently large even integers is the sum of a prime plus a number with no more than two prime factors (a P Goldbach conjecture also showed that every even number is the difference between a prime and a P Hardy and Wright give an argument for this conjecture in their well known footnote [HW79, p15] which goes roughly as follows. primes.utm.edu /notes/conjectures   (566 words)

 Goldbach's Conjecture   (Site not responding. Last check: 2007-10-31) In his famous letter to Leonhard Euler dated June 7th 1742, Christian Goldbach first conjectures that every number that is a sum of two primes can be written as a sum of "as many primes as one wants". On the margin of his letter, he then states his famous conjecture that every number is a sum of three primes: The ternary conjecture has been proved under the assumption of the truth of the generalized Riemann hypothesis and remains unproved unconditionally for only a finite (but yet not computationally coverable) set of numbers. www.mscs.dal.ca /~joerg/res/g-en.html   (390 words)

 The Poincare conjecture   (Site not responding. Last check: 2007-10-31) The Poincare Conjecture is essentially the first conjecture ever made in topology; it asserts that a 3-dimensional manifold is the same as the 3-dimensional sphere precisely when a certain algebraic condition is satisfied. The conjecture was formulated by Poincare around the turn of the 20th century. The Generalized Poincare Conjecture states that for every n, an n-dimensional manifold homotopy equivalent to the n-sphere is homeomorphic to the n-sphere. www.math.unl.edu /~mbrittenham2/ldt/poincare.html   (2822 words)

 Jacobian Conjecture One simple and useful result from the Jacobian conjecture is that by a simple reduction argument we know if there is a counter example to the Jacobian conjecture, then there must be a counter example \$f, g\$ with the degrees of \$f, g\$ non-divisible by each other. Abhyankar was one of the main movers of this conjecture and motivated research on the subject. This conjecture could be understood by anyone with a background in Calculus and hence it was studied by mathematicians in many disciplines, especially Algebra, Analysis, and Complex Geometry. www.math.purdue.edu /~ttm/jacobian.html   (859 words)

 Ivars Peterson's MathTrek - The Amazing ABC Conjecture That conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama. The key element appears to be a problem termed the ABC conjecture, which was formulated in the mid-1980s by Joseph Oesterle of the University of Paris VI and David W. Masser of the Mathematics Institute of the University of Basel in Switzerland. That conjecture offers a new way of expressing Diophantine problems, in effect translating an infinite number of Diophantine equations (including the equation of Fermat's last theorem) into a single mathematical statement. www.maa.org /mathland/mathtrek_12_8.html   (1179 words)

 Conjecture 33. The Goldbach Temptation After a little empirical verification of the conjecture, I suggested to Perry that the implicit "sufficiently even large number" might be the not so large but the nice number 210. The first one is a very ingenious modification made by Rodríguez to another conjecture stated by Euler. In later e-mails I clarified to him that this could be a sufficient condition for the Goldbach Conjecture. www.primepuzzles.net /conjectures/conj_033.htm   (849 words)

 Goldbach Conjecture Research This conjecture dates from 1742 and was discovered in correspondence between Goldbach and Euler. Goldbach made the conjecture that every odd number > 6 is equal to the sum of three primes. Euler replied that Goldbach's conjecture was equivalent to the statement that every even number > 4 is equal to the sum of two primes. www.petrospec-technologies.com /Herkommer/goldbach.htm   (1445 words)

 3x+1 conjecture verification results The 3x+1 conjecture [1], [2, problem E16] asserts that starting from any positive integer n the repeated iteration of T(x) eventually produces the integer 1, after which the iterates will alternate between the integers 1 and 2. In order to test the 3x+1 conjecture, in 1996 we wrote a computer program (in the programming language C), which computed the trajectories of all initial values of n smaller that a given limit and having a stopping time known to be larger than 40 [3]. Since no counter-example was found, the 3x+1 conjecture is probably true for all positive integers not larger than this verification limit. www.ieeta.pt /~tos/3x+1.html   (742 words)

 Goldbach conjecture verification The Goldbach conjecture is one of the oldest unsolved problems in number theory [1, problem C1]. 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) In this table [22k, compressed with gzip] we present all values of S(p) we were able to compute, as well as counts of the number of times each (small) prime was used in a minimal Goldbach partition. www.ieeta.pt /~tos/goldbach.html   (1292 words)

 Conjectures in Geometry: Triangle Sum Many students may already be familiar with this conjecture, which states that the angles in a triangle add up to 180 degrees. So this conjecture tells us that if we know two of the angles in a triangle, then we can find the third angle quite simply. Conjecture (Triangle Sum): The sum of the interior angles in any triangle is 180 degrees. www.geom.uiuc.edu /~dwiggins/conj04.html   (207 words)

 The Beal Conjecture   (Site not responding. Last check: 2007-10-31) The conjecture and prize was announced in the December 1997 issue of the Notices of the American Mathematical Society. Since that time Andy Beal has increased the amount of the prize for his conjecture. The prize is now this: \$100,000 for either a proof or a counterexample of his conjecture. www.math.unt.edu /~mauldin/beal.html   (274 words)

 Scientific American: Henri Poincar¿, His Conjecture, Copacabana and Higher Dimensions For 100 years mathematicians have been trying to prove a conjecture that was first proposed by Henri Poincar¿ relating to an object known as the three-dimensional sphere, or 3-sphere. Here I focus on Poincar¿ himself and the early years of his conjecture, in particular the astonishing results that proved higher-dimensional versions of the conjecture in the latter half of the twentieth century. Smale heard about the Poincar¿ conjecture in 1955, while he was a graduate student at the University of Michigan in Ann Arbor. www.sciam.com /print_version.cfm?articleID=0003848D-1C61-10C7-9C6183414B7F0000   (1582 words)

 Beal's Conjecture: A Search for Counterexamples The conjecture is obviously related to Fermat's Last Theorem, which was proved true by Andrew Wiles in 1994. A wide array of sophisticated mathematical techniques could be used in the attempt to prove the conjecture true (and the majority of mathematicians competent to judge seem to believe that it likely is true). But, I said, I did recall another contest related to Fermat's last Theorem, which would also be difficult to prove true, but which might be shown to be false with some adding and multiplying (of numbers that are a bit beyond the typical kindergarden range). www.norvig.com /beal.html   (1365 words)

 Xinhua - English Yau rated the conjecture as one of the major mathematical puzzles of the 20th Century. "The conjecture is that if in a closed three-dimensional space, any closed curves can shrink to a point continuously, this space can be deformed to a sphere," he said. By the end of the 1970s, U.S. mathematician William P. Thurston had produced partial proof of Poincar's Conjecture on geometric structure, and was awarded the Fields Prize for the achievement. news.xinhuanet.com /english/2006-06/04/content_4644754.htm   (548 words)

 Goldbach's Conjecture (II)   (Site not responding. Last check: 2007-10-31) This conjecture has not been proved nor refused yet. No one is sure whether this conjecture actually holds. The problem here is to write a program that reports the number of all the pairs of prime numbers satisfying the condition in the conjecture for a given even number. acm.uva.es /p/v6/686.html   (184 words)

 The New Yorker: PRINTABLES The conjecture was potentially important for scientists studying the largest known three-dimensional manifold: the universe. The implications of the conjectures for other disciplines may not be apparent for years, but for mathematicians the problems are fundamental. In 1976, he proved a twenty-year-old conjecture pertaining to a type of manifold that is now crucial to string theory. www.newyorker.com /printables/fact/060828fa_fact2   (9004 words)

 Lusztig's conjecture In an effort to verify (or disprove) the modular Lusztig conjecture for the prime 5 for SL5, Anders Buch started rewriting ``Dynkin'' in C in the fall of 94, beginning his graduate studies in Aarhus. In April 95 we were finally able to verify the Lusztig conjecture in the SL5, p =5 case using the super computer (SGI Power Challenge) at Aarhus Universitet. The conjecture alluded to in Andersen-Jantzen-Soergel (Asterisque 220) is for restricted weights and p greater than or equal to the Coxeter number. home.imf.au.dk /abuch/dynkin   (781 words)

 Mathematical mysteries: the Goldbach conjecture The conjecture says only that there is at least one, and has nothing to say about whether there may be more. He made his conjecture in a letter to Leonhard Euler, who at first treated the letter with some disdain, regarding the result as trivial. Goldbach's conjecture, however, remains unproved to this day. pass.maths.org.uk /issue2/xfile/index.html   (387 words)

 Goldbach's Conjecture   (Site not responding. Last check: 2007-10-31) Every number greater than 2 can be written as the sum of three prime numbers. Today it is still unproven whether the conjecture is right. Anyway, your task is now to verify Goldbach's conjecture as expressed by Euler for all even numbers less than a million. acm.uva.es /p/v5/543.html   (277 words)

 conjecture - Wiktionary A statement or idea which has not been proven, but is thought to be true; a guess. I explained it, but it is pure conjecture whether he understood, or not. to conjecture (third-person singular simple present conjectures, present participle conjecturing, simple past conjectured, past participle conjectured) en.wiktionary.org /wiki/conjecture   (82 words)

 Science News Online (7/24/99): The Honeycomb Conjecture The mathematicians' honeycomb conjecture therefore concerns a two-dimensional pattern-as if bees were creating a grid for laying out tiles to cover an infinitely wide bathroom floor. Fejes Tóth proved the honeycomb conjecture for the special case of filling the plane with any mixture of straight-sided polygons. In recent years, Morgan has refocused attention on the honeycomb conjecture and related questions, such as the most economical way of packaging a pair of identical volumes as double bubbles (SN: 8/12/95, p. www.sciencenews.org /sn_arc99/7_24_99/bob2.htm   (1749 words)

 Pitt math professor took best shot at cannonball conjecture   (Site not responding. Last check: 2007-10-31) But for four centuries, mathematicians had been unable to prove famed astronomer and mathematician Johannes Kepler's 1611 conjecture that the pyramid is the best way to stack cannonballs. The proof is about 300 pages long -- not counting 40,000 lines of computer code and three billion bytes of data necessary to solve the puzzle that left mathematicians scratching their scalps for centuries. Since proving the Kepler conjecture, he also solved the honeycomb conjecture by proving that the hexagon-shaped cells in honeycombs maximize area and minimize the amount of beeswax. www.post-gazette.com /pg/07016/754107-115.stm   (875 words)

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