41. Open Problems On Perfect Graphs Unsolved problems on perfect graphs. http://www.cs.concordia.ca/~chvatal/perfect/problems.html

PERFECT PROBLEMS Created on 22 August, 2000 Last updated on 5 July, 2006 In May 2002, the Strong Perfect Graph Conjecture became the Strong Perfect Graph Theorem Details are here. As a part of the 1992 1993 Special Year on Combinatorial Optimization at DIMACS ftp://dimacs.rutgers.edu/pub/perfect/problems.tex If you have information on progress towards solving these problems or complaints in case I did not give credit where credit was due or suggestions for problems to add, please, send them to me Related pages: Problems from the AIM Workshop on Perfect Graphs (compiled by Maria Chudnovsky) A bibliography on perfect graphs Home pages of people interested in perfect graphs This collection is written for people with at least a basic knowledge of perfect graphs. Uninformed neophytes may look up the missing definitions on the web in Alexander Schrijver's lecture notes or in Jerry Spinrad's draft of a book on efficient graph representations etc. or in MathWorld . Books on perfect graphs include M. C. Golumbic, Algorithmic graph theory and perfect graphs.

42. Open-Ended Math Problems Start your math problems! This site is for the specific purpose of preparing Middle School students for OPENENDED problem solving on standardized tests. http://sln.fi.edu/school/math2/index.html

Open-Ended Math Problems GET READY, GET SET... This site is for the specific purpose of preparing Middle School students for OPEN-ENDED problem solving on standardized tests. We have divided each month into the five strands from the Philadelphia math standards: Number Theory Measurement Geometry Patterns, Algebra, and Functions Data, Statistics, and Probability There are three levels of difficulty for each standard. We have written and chosen problems from different sources that lend themselves to more than one way of solving. It is our hope that if these are done on a consistent, weekly basis, the students will feel more confident and comfortable at test time. To better prepare for the tests, students should answer with a picture, diagram, or paragraph explaining the solution and how they determined their answers. We are including possible answers and rubrics for assessment Try these with your students. Let us know what you think and how they did. September Problems October Problems November Problems December Problems ... April Problems Gwenn Holtz (7th Grade) and Mary Lee Malen (6th Grade) For questions or comments, contact the

43. The Valuation Theory Home Page - Open Problems Open Problem 1 Generalize Abhyankar's Going Up and Coming Down for local uniformization to arbitrary finite transcendence degree. Possible applications To local http://math.usask.ca/fvk/Probl.html

The Valuation Theory Home Page Open Problems Open Problem 1: Generalize Abhyankar's "Going Up" and "Coming Down" for local uniformization to arbitrary finite transcendence degree. Possible applications: To local uniformization and resolution of singularities, in particular, in positive characteristic. Posted by F.-V. Kuhlmann on February 4, 1999 A valued field (K,v) is called spherically complete if every nest of balls has a non-empty intersection. This holds if and only if every Pseudo-Cauchy sequence in K has a pseudo limit in K. By the work of Kaplansky, this in turn holds if and only if the field is maximally valued, i.e., has no proper immediate extensions. Take a polynomial f with coefficients in K. The following is known: 1) If f is a polynomial in one variable, then the image f(K) is spherically complete, just as a set with the ultrametric induced by the valuation v. If f is an additive polynomial in several variables, then under a certain additional condition, f(K) is again spherically complete. Open Problem 2: Prove or disprove that f(K) is spherically complete for all additive polynomials in several variables.

44. UNSOLVED PROBLEMS AND REWARDS By Clark Kimberling. Offers prizes for solutions of some problems in number theory. http://faculty.evansville.edu/ck6/integer/unsolved.html

Unsolved Problems and Rewards Stated below are a few challenging problems. If you are first to publish a solution, let me know, and collect your reward! Or, if you find a short solution and you are quite sure it is correct and complete, send it to ck6@evansville.edu. If accepted, your proof will be published on this site - see, for example, Problem 8. 1. The Kolakoski Sequence: 122112122122112112212112122... This sequence is is identical to its own runlength sequence. Reward: $200.00 for publishing a solution of any one of the five problems stated in Integer Sequences and Arrays. The sequence originates in William Kolakoski , "Self generating runs, Problem 5304," American Mathematical Monthly 72 (1965) 674. For a proof that the Kolakoski sequence is not periodic, see the same Monthly See also Kolakoski Sequence at MathWorld 2. A Sequence Is every positive integer a term of this sequence: Reward: $300.00. To generate the sequence, visit Kimberling Sequence at MathWorld and Generator For a discussion and variant of the problem, see Richard K. Guy

45. Sensei's Library: Open Problems Sensei's Library, page Open Problems, keywords Theory. SL is a large WikiWikiWeb about the game of Go (Baduk, Weiqi). It's a collaboration and community site. Everyone can http://senseis.xmp.net/?OpenProblems

46. Unsolved Problems And Conjectures Regarding equal sums of like powers, compiled by Chen Shuwen. http://euler.free.fr/eslp/unsolve.htm

Equal Sums of Like Powers Unsolved Problems and Conjectures The Prouhet-Tarry-Escott Problem a k + a k + ... + a n k = b k + b k + ... + b n k k n Is it solvable in integers for any n Ideal solutions are known for n = 1, 2, 3, 4, 5, 6, 7, 8 ,9, 11 and no other integers so far. How to find new solutions for n = 10 and How to find the general solution for n The general solution is known for ( k = 1 ) ( k = 1, 2 ) ( k = 1, 2, 3 ) only. The complete symmetric solution have been found for ( k = 1, 2, 3, 4 ) How to find a new solution of the type ( k =1, 2, 3, 4, 5, 6, 7, 8 ) How to find non-symmetric ideal solutions of ( k =1, 2, 3, 4, 5, 6, 7, 8 ) and ( k =1, 2, 3, 4, 5, 6, 7, 8, 9 ) How to find a solution chain of the type ( k = 1, 2, 3, 4 ) a a a a a b b b b b c c c c c ( k = 1, 2, 3, 4 ) Solution chains are known for ( k = 1 ) ( k = 1, 2 ) ( k = 1, 2, 3 ) and ( k = 1, 2, 3, 4, 5 ) so far. Some other open problems are present on Questions by Lander-Parkin-Selfrige (1967) a k + a k + ... + a m k = b k + b k + ... + b n k Is ( k m n ) always solvable when m n k Is it true that ( k m n ) is never solvable when m n k For which k m n such that m n k is ( k m n ) solvable ?

47. Quantum Information: Problems Feb 16, 2007 Entangled Hbars. Open Problems. On this page we collect problems in Quantum Information Theory we or our contributors find worthy of http://www.imaph.tu-bs.de/qi/problems/

IMaPh Address Staff Research ... Quantum Information Open Problems On this page we collect problems in Quantum Information Theory we or our contributors find worthy of attention. The list is ordered by "last submission on top". Numbering is kept stable, so you can refer to the problems by number. Click on the title to find the statement of the problem and some information on known partial results. This information will be updated whenever significant progress is brought to our attention. To make this page interesting we obviously need input from the community, so please contribute good problems . By this we mean problems in Quantum Information Theory, which are stated in a self-contained way in the current terminology of the field, are open to the best of your knowledge, and pose an interesting challenge to other researchers. We will make an effort to publish all good problems quickly, but may reject contributions we find less suitable. Send contributions by email to . The format of contributions is free. Text written in simple LaTeX, and divided into sections such as Problem/ Background/ Partial Results/ Remarks/ Literature creates the smallest workload for us. contact date last progress solved by All the Bell Inequalities R.F. Werner

48. The RTA List Of Open Problems The RTA list of open problems summarizes open problems in the field of the International Conference on Rewriting Techniques and Applications (RTA). http://rtaloop.mancoosi.univ-paris-diderot.fr/

The RTA list of open problems The RTA list of open problems summarizes open problems in the field of the International Conference on Rewriting Techniques and Applications (RTA). For the RTA 2002 conference, the topics of RTA were given as Applications: case studies; rule-based programming; symbolic and algebraic computation; theorem proving; functional and logic programming; proof checking. Foundations: matching and unification; completion techniques; strategies; constraint solving; explicit substitutions. Frameworks: string, term, and graph rewriting; lambda-calculus and higher-order rewriting; conditional rewriting; proof nets; constrained rewriting and deduction; categorical and infinitary rewriting. Implementation: compilation techniques; parallel execution; rewriting tools. Semantics: equational logic; rewriting logic. The RTA list of open problems was created in 1991 by Nachum Dershowitz Jean-Pierre Jouannaud and Jan Willem Klop on occasion of the RTA’91 conference. Updated lists have since been published at RTA’93, RTA’95 and RTA’98. Since October 1997 the list of open problems is maintained as a web service. This effort is, at the moment, led by

49. 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

50. Open Problems In Group Theory We now have over 200 open problems in our collection, and we invite the mathematical community to submit more problems as well as comments, suggestions, http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html

OPEN PROBLEMS in combinatorial and geometric group theory Welcome to the new home of Open problems The original collection has been selected by G.Baumslag, A.Myasnikov and V.Shpilrain with the help of several other people. In particular, we are grateful to G.Bergman G.Conner W.Dicks R.Gilman ... I.Kapovich , V. Remeslennikov, V.Roman'kov E.Ventura and D.Wise for useful comments and discussions. We also acknowledge the NSF support through the grant DMS-0405105 that has helped us maintain this site. We now have over 200 open problems in our collection, and we invite the mathematical community to submit more problems as well as comments, suggestions, and/or criticism. Please send us e-mail at wwalgebra@yahoo.com HALL OF FAME We have arranged the problems under the following headings: OUTSTANDING PROBLEMS Auxiliary Problems FREE GROUPS ONE-RELATOR GROUPS ... GROUP ACTIONS since 10/10/02 FastCounter by bCentral

51. 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.

52. Papers On The 3x + 1 / 3n + 1 Problem, Fermat's Last Theorem, And Other Mathemat Papers by Peter Schorer describing several new approaches. http://www.occampress.com

Welcome to Occam Press! Information about Occam Press and about this web site. A note to professional mathematicians. A note to graduate students. The following papers, essays, and notes by Peter Schorer: Papers on the 3 x + 1 Problem (aka the 3n + 1 Problem, the Syracuse Problem, etc.) including: "A Solution to the 3x + 1 Problem" ... Essay, "Notes Toward a Pragmatics-Based Linguistics" William Curtis's book, How to Improve Your Math Grades , which sets forth a radical new organization of mathematical subjects aimed at improving the speed of problem solving. Paper, "Good Mathematical Writing Style: Summary of Rules" Information about Occam Press: Occam Press is a small publisher located in Berkeley, CA. It was created to provide an outlet for independent scholars, including mathematicians and computer scientists working outside the university. We will be placing entire works on this web site. Interested persons will be able to buy printed copies directly from us. However, until the works have been placed on the web site, we offer brief descriptions of each. Interested persons may obtain sample pages, and more information, by e-mailing or calling us, or by sending us surface mail. Occam Press 2538 Milvia St.

53. IPAM - Workshop I: Number Theory And Cryptography - Open Problems Securing Cyberspace Applications and Foundations of Cryptography and Computer Security. http://www.ipam.ucla.edu/programs/scws1/

Securing Cyberspace: Applications and Foundations of Cryptography and Computer Security Workshop I: Number Theory and Cryptography - Open Problems October 9 - 13, 2006 Schedule and Presentations Program Poster PDF Hotel Accommodation and Air Travel Organizing Committee: Arjen Lenstra, Chair (École Polytechnique Fédérale de Lausanne (EPFL)) Don Blasius (UCLA) Kristin Lauter (Microsoft Research) Alice Silverberg (University of California, Irvine) Joseph Silverman (Brown University) Scientific Background Cryptography depends on a continuing stream of new insights and methods from number theory, arithmetic algebraic geometry, and other branches of algebra. In the past, there have been important developments in primality testing, factoring large integers, lattice-based cryptography, sieve methods, elliptic curve cryptography, ECPP, torus-based cryptosystems, discrete log problems, Weil pairing, cyclicity of elliptic curves and hyperelliptic cryptosystems. The content of this workshop will be based on emerging developments and discussion of open problems posed by applications. Invited Speakers Don Blasius (UCLA) Johannes Buchmann (Technishche Universtitat Darmstadt) Denis Charles (Microsoft Research) Jean-Marc Couveignes (Université de Toulouse II (Le Mirail)) Yvo Desmedt (University College London) Kirsten Eisentrager (University of Michigan)

54. The Generalised 3x+1 Problem A survey by Keith Matthews. http://www.numbertheory.org/pdfs/survey.pdf

55. Onezero » An Image From The Collatz Problem By Andrew Shapira. The intensity of a point denotes the time taken to terminate. http://www.onezero.org/collatz-image

onezero Printer-friendly page - just print An Image From the Collatz Problem Andrew Shapira February 15, 1998 Includes minor subsequent revisions such as web link updates. Introduction Consider the following rule that maps a given positive integer n to another: if n is even, the next integer is n/2 ; if n is odd, the next integer is . Starting at an arbitrary integer, we can repeatedly apply the rule to obtain a sequence of integers. For example: 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. . (See the table of contents at the sci.math FAQ One day, Roddy Collins was showing me the Fractint package. Fractint is a package for generating images of fractals and fractal-like structures. Fractint has its own programming language, as well as a huge number of options for doing things like manipulating images and controlling parameters. The main operation in the programming language is to repeat a certain region of code until some termination condition is reached. The color or intensity at a given pixel corresponds to how many times the loop was iterated for the object that corresponds to the pixel. This reminded me of the Collatz problem, and I wondered whether we could use Fractint to draw a picture of the Collatz problem. I thought it would be neat to use the same kind of spiral pattern that has sometimes been used to graphically display prime numbers:

56. Mesh Generation: Open Problems ICS 280G, Spring 1997 Mesh Generation for Graphics and Scientific Computation Open Problems. We proved (4/3/97) the existence of triangulations of any polygon or straight line http://www.ics.uci.edu/~eppstein/280g/open.html

ICS 280G, Spring 1997: Mesh Generation for Graphics and Scientific Computation Open Problems We proved ( ) the existence of triangulations of any polygon or straight line graph, and of convex quadrilateralizations of any orthogonal polygon. What about curved objects? Do spline-polygons have spline-triangulations? An example formed by connecting four quarter-ellipses shows Steiner points may be needed, even for quadratic splines, but maybe they only need to be added in the interior of the splinegon. On we went over dynamic programming techniques for optimal triangulation (e.g. minimum total edge length) of simple polygons, in O(n ) time or O(E ) if the visibility graph has E edges. So the slowest case is seemingly the most simple, when the polygon is convex. Can we find the minimum length triangulation of convex polygons in o(n ) time? Steve S. suggested Frances Yao's generalization of Knuth's speedup to optimal binary search tree construction (which has the same general dynamic programming form) but it doesn't seem to work. The same dynamic programming methods also work for optimal quadrilateralization, in time O(n

57. Lychrel Number -- From Wolfram MathWorld MathWorld article and resources. http://mathworld.wolfram.com/LychrelNumber.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Sequences Lychrel Number The first few numbers not known to produce palindromes when applying the 196-algorithm (i.e., a reverse-then-add sequence ) are sometimes known as Lychrel numbers. This term was coined by VanLandingham in 2002 as a variation of his girlfriend Cheryl's name (VanLandingham, pers. comm., Jun. 9, 2005). The first few Lychrel numbers are 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, ... (Sloane's ). All Lychrel numbers with 17 or fewer digits are tabulated at http://www.p196.org/ SEE ALSO: 196-Algorithm Reverse-Then-Add Sequence REFERENCES: Sloane, N. J. A. Sequence in "The On-Line Encyclopedia of Integer Sequences." VanLandingham, W. "196 and Other Lychrel Numbers." http://www.p196.org/ VanLandingham, W. "Blackboard Archive." May 13, 2002. http://home.cfl.rr.com/p196/archive.html CITE THIS AS: Weisstein, Eric W. "Lychrel Number." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/LychrelNumber.html

58. Open Problems Of Paul Erd Os In Graph Theory File Format PDF/Adobe Acrobat Quick View http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.7532&rep=rep1&am

59. Open Problems Open Problems There are a variety of natural questions left unanswered by the work described. Below, we list a few natural directions for further work in this area. http://www.usenix.org/publications/library/proceedings/ec98/full_papers/harkavy/

Next: References Up: Electronic Auctions with Private Previous: Error analysis Open Problems There are a variety of natural questions left unanswered by the work described. Below, we list a few natural directions for further work in this area. Work is under way on variations on the protocol which will address tie-breaking issues, reduce communication costs, and Tie-breaking The protocol described does not provide efficient tie-breaking without some loss of privacy. Communication Costs While we believe the communication costs are sufficiently small to make this protocol practical in many situations, low-value auctions are likely to play an increasingly important role in electronic commerce. Efficiency improvements that enable auctions with private bids in these low-value situations could be very useful. Hierarchical Auctions In some situations, it might be advantageous to hold sub-auctions which partially determine the outcome of the auction. For example, each country might hold a sub-auction, with the leading candidates from each country participating in a final auction. In order to preserve privacy, the winners of the sub-auctions and their bids must not be revealed. Double Auctions and Auction Markets A double auction is a more general form of auction where there are multiple sellers and multiple buyers. All parties tender bids and a market clearing price is determined from those bids. A market clearing price is the equilibrium price at which the supply and demand (in units of the good) are equal. An auction market is a generalization of a double auction to continuous time. New bids are added and removed over time, causing the market clearing price to fluctuate. The stock market is a well known example of an auction market. Double auctions and auction markets are powerful market mechanisms, and privacy protecting protocols for these mechanism would be desirable. However, to be useful, such protocols must be highly efficient, particularly in the case of auction markets.

60. Three Years Of Computing Palindrome Quest. Reporting computations on the 196 problem. http://www.fourmilab.ch/documents/threeyears/threeyears.html

Three Years Of Computing Final Report On The Palindrome Quest by John Walker May 25th, 1990 Pick a number. Reverse its digits and add the resulting number to the original number. If the result isn't a palindrome, repeat the process. Do all numbers in base 10 eventually become palindromes through this process? Nobody knows. For example, start with 87. Applying this process, we obtain: 87 + 78 = 165 165 + 561 = 726 726 + 627 = 1353 1353 + 3531 = 4884, a palindrome In order for addition of a digits-reversed number to yield a palindrome, there must be no carries in the addition and hence each pair of digits must sum to 9 or less. Whether all numbers eventually become palindromic under this process is unproved, but all numbers less than 10,000 have been tested. Every one becomes a palindrome in a relatively small number of steps (of the 900 3-digit numbers, 90 are palindromes to start with and 735 of the remainder take less than 5 reversals and additions to yield a palindrome). Except, that is, for 196. This number had been carried through 50,000 reversals and additions by P. C. Leyland, yielding a number of more than 26,000 digits without producing a palindrome. Later, P. Anderton continued the process up to 70,928 digits without encountering a palindrome. On August 12, 1987, I put my

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