121. 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

122. Conferences, Department Of Mathematics, Texas A&M University Problems taken from workshop lectures given at Texas A M University. http://www.math.tamu.edu/conferences/linanalysis/problems.html

Courses Undergraduate Graduate Positions Available ... Contact Us Open Problems in Linear Analysis and Probability The problems here were either submitted specifically for the purpose of inclusion in this page, or were taken from talks given during the Concentration Week on Metric Geometry and Geometric Embeddings of Discrete Metric Spaces. Click here to view the PDF version of the file. Click here to view open problems list on embeddings of finite metric spaces compiled by Jiri Matousek Problem #1 (Submitted by Guoliang Yu) Problem #2 (Submitted by Guoliang Yu) Problem #3 (Submitted by Yuval Peres) Problem #4 (Submitted by Yuval Peres) Do planar graphs (with graph distance) have a Markov type 2. Problem #5 (Submitted by Moses Charikar) How well do $l_2^2$ (negative type) metrics embed into $l_1$? Problem #6 (Submitted by Sanjeev Arora) Problem #7 (Submitted by Bruce Kleiner) Characterize metric spaces which bi-Lipschitz embed into $L^2$. Problem #8 (Submitted by Leonid Kovalev) $f(X)$ is doubling Problem #9 (Submitted by Piotr Nowak) Do expanders (with constant degree) embed coarsely into any uniformly convex Banach space?

123. MathWorld News: Draft Proof Of Catalan's Conjecture Circulated News brief announcing the break through that provides a history and description of the equation. http://mathworld.wolfram.com/news/2002-05-05/catalan/

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Headline News Draft Proof of Catalan's Conjecture Circulated By Eric W. Weisstein May 5, 2002Today, in an email sent to the NMBRTHRY mailing list, number theorist Alf van der Poorten confirmed that an apparent proof of the long-outstanding Catalan's conjecture has been circulated to a group of mathematicians by number theorist Preda Mihailescu. The conjecture in question was made by Belgian mathematician in 1844, and it states that 8 and 9 (2 and 3 ) are the only consecutive powers excluding and 1. In other words, Catalan conjectured that is the only nontrivial solution to the so-called Catalan's Diophantine problem x p y q three consecutive powers existed (Ribenboim 1996), Catalan's conjecture itself has stubbornly refused attack for more than a century and a half. The first groundbreaking result was that of Robert Tijdeman (1976), who showed that there can be at most a finite number of exceptions should the conjecture not hold. This led to considerable computational efforts, and in 1999, Maurice Mignotte showed that if a nontrivial solution exists, then 10 p and 10 q (Peterson 2000).

124. Open Problems From CCCG 2007 File Format PDF/Adobe Acrobat Quick View http://cccg.ca/proceedings/2008/paper45.pdf

125. 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:

126. Electric Grid Control Algorithms Open Problems File Format PDF/Adobe Acrobat Quick View http://cnls.lanl.gov/~chertkov/SmarterGrids/Talks/Abdallah.pdf

127. A New Extreme ABC-example Explanation of an example with quality 1.920859, found by Benne de Weger and Niklas Broberg. http://deweger.xs4all.nl/oud/broberg.txt

A new extreme abc-example ========================= For coprime integers a, b, c such that a + b = c the well known "abc-conjecture" says that H(a,b,c)

128. 21 Open Problems In Artificial Intelligence Dec 19, 2007 Peter has come up with a list of 21 (important) open problems in the field of Artificial Intelligence. I am not aware of any such list http://www.daniel-lemire.com/blog/archives/2007/12/19/21-open-problems-in-artifi

@import url( http://www.daniel-lemire.com/blog/wp-content/themes/lemiretheme/style.css ); 21 open problems in Artificial Intelligence Peter has come up with a list of 21 (important) open problems in the field of Artificial Intelligence . I am not aware of any such list anywhere, so this might be an important contribution. For comparison, Wikipedia as a list of open problems in Computer Science . In the field of database, the closest thing to a list of open problems would be the Lowell report : it falls short of providing true open problems however. I am a bit surprised to see Learning Chess , but not Learning Go , on his list since I have the impression that Deep Blue has pretty much learned to play Chess at a very high level, whereas the same is not true of Go Self-References in software . I am no expert in AI, but it seems to me that the main mystery we are facing today, the deepest mystery of all, is what is consciousness Some say that as computers grow larger, more connected and more powerful, they will acquire consciousness. Maybe. I believe however that consciousness has to do with software that can reference and modify itself. Note that we can figure out what consciousness is without building a conscious robot, and we may build a conscious robot without knowing what consciousness is. Approximate queries in databases. As we now have

129. Collatz 3n+1 Problem Structure Observations posted by Ken Conrow to stimulate further research. http://www-personal.ksu.edu/~kconrow/

Ken Conrow Home Page Collatz 3n+1 Problem Structure Mathematicians who refer to the problem as the problem were never brainwashed by FORTRAN (as I was) into the belief that n , not x , stands for an integer. This work has been ongoing for several years, and fragments of earlier approaches appear on these pages. An attempt has been made to label these fragments, and they are referenced among the later entries of the table of contents, below. Currently, I've obtained a detailed mapping of the residue sets which constitute the abstract Collatz predecessor tree. Systematic features of this structure permit development of formulas whose infinite summation indicates the presence of all the odd positive integers therein. The recursive program available I hope someone who can formalize mathematical proofs will see the potential here and take the appropriate set of ideas and sketch or complete a formal proof of the conjecture using them. You may communicate with me by e-mail at kconrow@ksu.edu . Reports of errors and constructive comments will be particularly welcome. Ideas Basic to the Structural View of the Collatz Graph n +1 Problem Statement and References n ... +1 Predecessor Tree, Three Views

130. Collatz Problem -- From Wolfram MathWorld From Eric Weissten s World of Mathematics. Article with references and links. http://mathworld.wolfram.com/CollatzProblem.html

131. International Conference On The Collatz Problem Katholische Universit t Eichst tt, Germany; 56 August 1999. On-line proceedings and group photo. http://www.math.grin.edu/~chamberl/conf.html

International Conference on the Collatz Problem and Related Topics August 5-6, 1999 This conference is intended for anyone interested in the 3x+1 problem ( also known as the Syracuse algorithm, Collatz', Kakutani's, or Ulam's problem), and related mathematics. CONFERENCE SCHEDULE CONFERENCE PROCEEDINGS E-mail: xhillner@aol.com Phone: (08421) 982010 Fax : (08421) 982080 You may also want to see other places of accomodation ; click on the word "Tourist Info" and then "Hotels". REGISTRATION: US$60 or 54 Euro, payable at the conference. FINANCIAL SUPPORT: A limited amount of financial support may be available. The Willibaldsburg (castle) St. Peter's Dominican Church ORGANIZERS: Marc Chamberland Department of Mathematics Grinnell College Grinnell, Iowa 50112 U.S.A. Office: (515) 269-4207 Fax: (515) 269-4984 chamberl@math.grin.edu Germany Telefon: (08421) 93-1456 Telefax: (08421) 93-1789 guenther.wirsching@ku-eichstaett.de

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