101. PlanetMath: Weil Conjectures Mar 3, 2006 conjectures made by Weil on the form of the zeta function of a variety over a finite field. Specifically he thought it should be rational, http://planetmath.org/encyclopedia/WeilConjectures.html

(more info) Math for the people, by the people. donor list find out how Encyclopedia Requests ... Advanced search Login create new user name: pass: forget your password? Main Menu sections Encyclopædia 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 Weil conjectures (Conjecture) Conjectures made by Weil on the form of the zeta function of a variety over a finite field . Specifically he thought it should be rational , it should split into polynomial parts with integral coefficients , with roots of a certain magnitude, and of degree Betti number . Further there should be a functional equation (Flesh this out ;)) "Weil conjectures" is owned by full author list view preamble get metadata View style: jsMath HTML HTML with images page images TeX source Log in to rate this entry. view current ratings Cross-references: functional equation Betti number degree roots ... conjectures There are 7 references to this entry. This is version 3 of Weil conjectures , born on 2006-03-03, modified 2008-06-09.

102. Math 608R: Etale Cohomology And The Weil Conjectures The conjectures of André Weil have influenced (or directed) much of 20th led to the formulation and the proof of the Weil conjectures....... http://www.math.umd.edu/~atma/Math608R.htm

Math 608R: Etale Cohomology and the Weil conjectures Fall 2007 MWF 1pm -1:50pm, PHY4208 Professor Niranjan Ramachandran, 4115, x5-5080 Textbooks: (MEC) Milne’s notes on “ Etale Cohomology supplemented by (ECW) E. Freitag and R. Kiehl Etale Cohomology and the Weil conjectures available online in djvu format) ( BAMS Review by N. Katz) Description: The conjectures of Andr Weil have influenced (or directed) much of 20th century algebraic geometry. These conjectures generalize the Riemann hypothesis (RH) for function fields (alias curves over finite fields), conjectured and verified in some special cases) by Emil Artin . Helmut Hasse proved RH for elliptic function fields. RH for general function fields was finally proved by Weil who then formulated his conjectures for higher dimensional algebraic varieties over finite fields. The last of this set of conjectures directly generalizes RH. The Weil conjectures are now known to be true (by work of Alexandre Grothendieck , Michael Artin Pierre Deligne , et al). This course will provide an overview of the methods and ideas which have led to the formulation and the proof of the Weil conjectures.

103. Conjecture - Wikipedia, The Free Encyclopedia A conjecture is a proposition that is unproven but appears correct and has not been disproven. Karl Popper pioneered the use of the term conjecture in http://en.wikipedia.org/wiki/Conjecture

Conjecture From Wikipedia, the free encyclopedia Jump to: navigation search This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations Please improve this article by introducing more precise citations where appropriate (August 2010) A conjecture is a proposition that is unproven but appears correct and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy . Conjecture is contrasted by hypothesis (hence theory axiom principle ), which is a testable statement based on accepted grounds. In mathematics , a conjecture is an unproven proposition or theorem that appears correct. Contents Famous conjectures Counterexamples Use of conjectures in conditional proofs Undecidable conjectures ... edit Famous conjectures Until recently, the most famous conjecture was Fermat's Last Theorem citation needed The conjecture taunted mathematicians for over three centuries before Andrew Wiles , a Princeton University research mathematician, finally proved it in 1995, and now it may properly be called a theorem. Other famous conjectures include: The abc conjecture P NP The Collatz conjecture Beal's conjecture The Poincaré theorem (proven by Grigori Perelman The Goldbach's conjecture The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e.g.

104. Goldbach Conjecture -- From Wolfram MathWorld Article from MathWorld. http://mathworld.wolfram.com/GoldbachConjecture.html

105. Several Proofs Of The Twin Primes Conjecture Goldbach s Conjecture proves and extends the Twin Primes Conjecture as probable. http://www.coolissues.com/mathematics/Tprimes/tprimes.htm

SEVERAL PROOFS OF THE TWIN PRIMES AND GOLDBACH CONJECTURES James Constant math@coolissues.com Proof of Goldbach's Conjecture, the Prime Number Theorem, and Euclid's Logic Provide Proofs of the Twin Primes Conjecture. Proof of the Twin Primes Conjecture Provides Proof of Goldbach's Conjecture Theorem There are infinitely many twin primes. Proof of the Twin Primes Conjecture Using Proofs of Goldbach's Conjecture or Using the Prime Number Theorem The twin primes conjecture (TPC) suggests that there is an infinite number of primes a and b with a difference , i.e., a - b = 2. Goldbach's conjecture (GC) suggests that every even number greater than is the sum s of two prime numbers a and b , i.e., a + b = s where s is even GC is proved by the author herein below and elsewhere For prime numbers a,b,c a - b = (a + c) - (b + c) even integer and thus, generally, a - b = 2k k = integer and since a + b is an even number a + b = 2n Now, using (2) and (3) results in a = n + k and b = n - k which say that for every single value of k primes a and b are separated by an interval and occur as numbers n + k and n - k . Suppose that n ,n ,n , . . . ,n

106. Ivars Peterson's MathTrek - The Amazing ABC Conjecture Article by Ivars Peterson. http://www.maa.org/mathland/mathtrek_12_8.html

Ivars Peterson's MathTrek December 8, 1997 The Amazing ABC Conjecture In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer. Fermat's last theorem, for instance, involves an equation of the form x n y n z n . More than 300 years ago, Pierre de Fermat (1601-1665) conjectured that the equation has no solution if x y , and z are all positive integers and n is a whole number greater than 2. Andrew J. Wiles of Princeton University finally proved Fermat's conjecture in 1994. In order to prove the theorem, Wiles had to draw on and extend several ideas at the core of modern mathematics. In particular, he tackled the Shimura-Taniyama-Weil conjecture, which provides links between the branches of mathematics known as algebraic geometry and complex analysis. That conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama. Goro Shimura of Princeton and Andre Weil of the Institute for Advanced Study provided key insights in formulating the conjecture, which proposes a special kind of equivalence between the mathematics of objects called elliptic curves and the mathematics of certain motions in space. The equation of Fermat's last theorem is one example of a type known as a Diophantine equation an algebraic expression of several variables whose solutions are required to be rational numbers (either whole numbers or fractions, which are ratios of whole numbers). These equations are named for the mathematician Diophantus of Alexandria, who discussed such problems in his book

107. The Abc Conjecture Maintained by Abderrahmane Nitaj. http://www.math.unicaen.fr/~nitaj/abc.html

THE ABC CONJECTURE HOME PAGE La conjecture abc est aussi difficile que la conjecture ... xyz. (P. Ribenboim) (read the story) The abc conjecture is the most important unsolved problem in diophantine analysis. (D. Goldfeld) Created and maintained by Abderrahmane Nitaj Last updated May 27, 2010 Index The abc conjecture Generalizations ... examples New hunters of abc examples rekenmeemetabc abc@home Largest good abc ... Contact The abc conjecture rad(n) For a natural number, let rad(n) be the product of all distinct prime divisors of n . E.g. if n then rad(n) The abc conjecture: Given any c C (rad(abc)) The abc conjecture was first formulated by Joseph Oesterlé [Oe] and David Masser [Mas] in 1985. Although the abc conjecture seems completely out of reach, there are some results towards the truth of this conjecture. 1986, C.L. Stewart and R. Tijdeman [Ste-Ti]: rad(abc) 1991, C.L. Stewart and Kunrui Yu [Ste-Yu1]: rad(abc) where C is an absolute constant, C and C are positive effectivley computable constants in terms of K. Gyory new results on the abc conjecture: To Index Generalizations The abc theorem for polynomials.

108. Introduction To The ABC Conjecture Slides by Barry Mazur. http://msri.org/publications/ln/hosted/ucb/1998/mazur/1/

109. XGC - An EXtension Of The Goldbach Conjecture An eXtension of the Goldbach Conjecture. Mathematica code. http://members.tripod.com/~aercolino/goldbach/

However, go to http://members.tripod.com/~aercolino/goldbach/xgc_printable.html for a single page version. Build your own FREE website at Tripod.com Share: Facebook Twitter Digg reddit document.write(lycos_ad['leaderboard']); document.write(lycos_ad['leaderboard2']);

110. Primesbehaviour, Primesbehaviour, Nature Template - PC Word 97 An insight into the Goldbach Conjecture. http://eureka.ya.com/angelgalicia30/Primesbehaviour.htm

111. THE STEPLADDER PROOF OF THE GOLDBACH CONJECTURE A proposed proof offered for criticism. http://gmazur.freeyellow.com/Goldbach.htm

THE STEPLADDER PROOF OF THE GOLDBACH CONJECTURE Version: 10/7/04 by Gregory Mazur ( greg@tell-all.com The Goldbach Conjecture: Every even number may be written as the sum of two odd primes. Assume even integer C is a counter-example to the Goldbach conjecture. A fixed sequence of primes, p_1 thru p_n, are each less than C. Given our assumption, there is a matched set of odd composites, r_1 thru r_n, such that C = p_i + r_i for n matched sets. Let p_1 = 3; C = p_1 + r_1; Also, C = p_n + r_n Let g1 be the first prime gap. Thus g1 = (5 – 3) = 2 C = (p_1 + g1) + (p_n + r_n – g1 – p_1) C = p_2 + (p_n + r_n – p_2) the next calculated smaller prime but not the next sequentially smaller prime as we step down from p_n to p_1. To favor the counter-example case, assume that we can encounter subsequent s at the same rate as r_2 thru r_(n/2). Also assume that we can encounter each sequentially smaller prime, p_n-i . Since we can construct every prime starting with p_1, neither of these two false assumptions are material to the outcome. On the contrary, the more primes, p_n-i that fail our test, the greater our confidence in the existence of a counter-example.

112. Poincaré Conjecture -- From Wolfram MathWorld A widely accessible statement of the Poincar Conjecture and its implications, with various links and references for further inquiry. http://mathworld.wolfram.com/PoincareConjecture.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) , where a three-sphere is simply a generalization of the usual sphere to one dimension compact manifold is homotopy -equivalent to the -sphere iff it is homeomorphic to the sphere . The generalized statement reduces to the original conjecture for topology of manifolds Whitehead link ) to his own theorem. The case of the generalized conjecture is trivial, the case is classical (and was known to 19th century mathematicians), (the original conjecture) appears to have been proved by recent work by G. Perelman (although the proof has not yet been fully verified), was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal was demonstrated by Zeeman (1961), was established by Stallings (1962), and was shown by Smale in 1961 (although Smale subsequently extended his proof to include all The Clay Mathematics Institute included the conjecture on its list of $1 million prize problems. In April 2002, M. J. Dunwoody produced a five-page paper that purports to prove the conjecture. However, Dunwoody's manuscript was quickly found to be fundamentally flawed (Weisstein 2002). A much more promising result has been reported by Perelman (2002, 2003; Robinson 2003). Perelman's work appears to establish a more general result known as the Thurston's geometrization conjecture , from which the Poincaré conjecture immediately follows (Weisstein 2003). Mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any substantial mistakes (Robinson 2003, Collins 2004). In fact, Collins (2004) goes so far as to state, "everyone expects [that] Perelman's proof is correct."

113. The Kepler Conjecture Information on the recent proof of Kepler conjecture on sphere packings. http://www.math.pitt.edu/~thales/kepler98/

This page is available for historical purposes only. It is a copy from www.math.lsa.umich.edu/~hales/countdown. It has not been maintained since 1998.

114. Catalan's Conjecture -- From Wolfram MathWorld Features an explanation of the equation and offers bibliographic references. http://mathworld.wolfram.com/CatalansConjecture.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Pegg Catalan's Conjecture and ) are the only consecutive powers (excluding and 1). In other words, is the only nontrivial solution to Catalan's Diophantine problem The special case and is the case of a Mordell curve Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288-1344) had already noted that the only powers of 2 and 3 that apparently differed by 1 were and (Peterson 2000). three consecutive powers exist (Ribenboim 1996), and it was also known that 8 and 9 are the only consecutive cubic and square numbers Tijdeman (1976) showed that there can be only a finite number of exceptions should the conjecture not hold. More recent progress showed the problem to be decidable in a finite (but more than astronomical) number of steps and that, in particular, if and are powers , then (Guy 1994, p. 155). In 1999, M. Mignotte showed that if a nontrivial solution exists, then (Peterson 2000). It had also been known that if additional solutions to the equation exist

115. Catalan's Conjecture - Wikipedia, The Free Encyclopedia Brief article offers a description of the problem and features links. http://en.wikipedia.org/wiki/Catalan's_conjecture

Catalan's conjecture From Wikipedia, the free encyclopedia Jump to: navigation search Catalan's conjecture (occasionally now referred to as Mihăilescu's theorem ) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu To understand the conjecture, notice that 2 and 3 are two powers of natural numbers , whose values 8 and 9 respectively are consecutive. The conjecture states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers of x a y b for x a y b x a y b Contents History Pillai's conjecture See also References ... edit History The history of the problem dates back at least to Gersonides , who proved a special case of the conjecture in 1343 where x and y were restricted to be 2 or 3. In 1974, Robert Tijdeman applied methods from the theory of transcendental numbers to show that there is an effectively computable constant C so that the exponents of all consecutive powers are less than C. As the results of a number of other mathematicians collectively had established a bound for the base dependent only on the exponents, this resolved Catalan's conjecture for all but a finite number of cases. However, the finite calculation required to complete the proof of the theorem was nonetheless too time-consuming to perform. Catalan's conjecture was proved by Preda Mihăilescu in April 2002, so it is now sometimes called

116. MUG ABC-conjecture Maple code to illustrate the conjecture discussed and refined. http://www.math.rwth-aachen.de/mapleAnswers/html/180.html

Maple User Group Answers [Anfang] [Hauptseite] [Suchen] ... [LDFM] abc-conjecture (12.9.96) John B. Cosgrave In the coming academic year I want to try to explain to my students the so-called 'abc-conjecture', a conjecture (initially by Oesterle', and refined by Masser) of the past decade which is rightly considered to be a truly profound one, with many deep consequences (see - for example - Dorian Goldfeld's article "Beyond the Last Theorem" in the March/April '96 issue of "The Sciences" (New York), or the article by Fields medallist Alan Baker in the recent centennial issue of the 'Mathematical Gazette'). Because of the help that I got in connection with my square-free question, and the related 'product' question, I am now in a position to experiment with the following programme, whose 'meaning' is simply this: one is trying to find relatively prime values of 'a' and 'b' (i.e. igcd(a, b)=1) such that their sum 'c' has the PROPERTY that the square-free part of a*b*c DIVIDED by 'c' has a 'small' value (in the procedure below, 'small' means 'less than 1'): Comment One: My computing facilities don't allow me to let 'n' get too large. n=500 took 1452 seconds on my 486. The last of the outputs - for n=500 - was:

117. Goldbach Conjecture Research Information on research and computations on the Goldbach Conjecture. By Mark Herkommer. http://www.petrospec-technologies.com/Herkommer/goldbach.htm

Goldbach Conjecture Research by Mark Herkommer May 24, 2004 The Conjecture... This conjecture dates from 1742 and was discovered in correspondence between Goldbach and Euler. It falls under the general heading of partitioning problems in additive number theory. 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. Because proving the second implies the first, but not the converse, most attention has been focused on the second representation. The smallest numbers can be verified easily by hand: Of course all the examples in the world do not a proof make. Research On The Conjecture... As a partitioning problem it is worth noting that as the numbers get larger the number of representations grows as well: This would suggest that the likelihood of finding that exceptional even number that is not the sum of two primes diminishes as one searches in ever larger even numbers. Euler was convinced that Goldbach's conjecture was true but was unable to find any proof (Ore, 1948). The first conjecture has been proved for sufficiently large odd numbers by Hardy and Littlewood (1923) using an "asymptotic" proof. They proved that there exists an n0 such that every odd number n > n0 is the sum of three primes. In 1937 the Russian mathematician Vingradov (1937, 1954) again proved the first conjecture for a sufficiently large, (but indeterminate) odd numbers using analytic methods. Calculations of n0 suggest a value of 3^3^15, a number having 6,846,169 digits (Ribenboim, 1988, 1995a).

118. The Prime Glossary: Goldbach's Conjecture From the Primes Glossary. Historical references and links. http://primes.utm.edu/glossary/page.php?sort=GoldbachConjecture

119. Swett, Research, Erdos-Strauss Conj The page establishes that the conjecture is true for all integers. Tables and software by Allan Swett. http://math.uindy.edu/swett/esc.htm

Allan Swett, Current Research on ESC... rev. 10/28/99 The Erdos-Strauss Conjecture 4/n = 1/a + 1/b + 1/c, (Note: Some of the linked pages are under revision.) Principal Ideas, Part 1: A C++ Program One may establish ESC(n) for a particular class of integers n using an identity. For example, the identity 4/(2+3x) = 1/(2+3x) + 1/(1+x) + 1/((1+x)(2+3x)) establishes that ESC(n) is true for n = 2, 5, 8, 11, 14, 17, ... (simply take x = 0, 1, 2, 3, 4, 5, ...). That is, ESC(n) is true for all positive integers n which are congruent to 2, mod 3. (Two integers are said to be "congruent mod n", if and only if n divides their difference. We abbreviate the fact that A and B are congruent mod n as A == B (mod n) .) A "generic" identity provides a set of similar results: Vocabulary: S(n) AX + W == (mod 4n-1) for some positive integer divisors X and W of n. Let E(n) denote the set of positive integers congruent, mod 4n-1, to an element of S(n). Then Theorem: Link to a proof of the Theorem.

120. Index The official Beal Conjecture site with information and links regarding the problem. http://www.bealconjecture.com

The Beal Conjecture Background Mathematicians have long been intrigued by Pierre Fermat's famous assertion that A x + B x = C x is impossible (as stipulated) and the remark written in the margin of his book that he had a demonstration or "proof". This became known as Fermat's Last Theorem (FLT) despite the lack of a proof. Andrew Wiles proved the relationship in 1994, though everyone agrees that Fermat's proof could not possibly have been the proof discovered by Wiles. Number theorists remain divided when speculating over whether or not Fermat actually had a proof, or whether he was mistaken. This mystery remains unanswered though the prevailing wisdom is that Fermat was mistaken. This conclusion is based on the fact that thousands of mathematicians have cumulatively spent many millions of hours over the past 350 years searching unsuccessfully for such a proof. It is easy to see that if A x + B x = C x then either A, B, and C are co-prime or, if not co-prime that any common factor could be divided out of each term until the equation existed with co-prime bases. (Co-prime is synonymous with pairwise relatively prime and means that in a given set of numbers, no two of the numbers share a common factor.) You could then restate FLT by saying that A x + B x = C x is impossible with co-prime bases. (Yes, it is also impossible without co-prime bases, but non co-prime bases can only exist as a consequence of co-prime bases.)

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