41. Home Page Of Keith Matthews University of Queensland (emeritus). Computational problems, Diophantine equations, LLL, Collatz problem. Originator and maintainer of the Number Theory Web. http://www.maths.uq.edu.au/~krm/

42. Collatz Collatz Problem A problem posed by L. Collatz in 1937, also called the 3 x +1 maping, Hase's algorithm, kakutani's problem, Syracse algorithm, Syracuseproblem http://library.thinkquest.org/C006364/ENGLISH/problem/Collatz.htm

Collatz Problem A problem posed by L. Collatz in 1937, also called the 3 x be an integer. Then the Collatz problem asks if iterating always returns to 1 for positive . This question has been tested and found to be true for all numbers (Oliveira e Silva 1999), improving the earlier results of (Vardi 1991, p. 129) and (Leavens and Vermeulen 1992) . The members of the sequence produced by the Collatz are sometimes known as hailstone numbers. Because of the difficulty in solving this problem, Erdos commented that "mathematics is not yet ready for such problems" (Lagarias 1985). If negative numbers are included, there are four known cycles (excluding the trivial cycle): (4, 2, 1), ( ), and ( ). The number of tripling steps needed to reach 1 for , 2, ... are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane's The Collatz problem was modified by Terras (1976, 1979), who asked if iterating always returns to 1 for initial integer value . If negative numbers are included, there are 4 known cycles: (1, 2), (

43. The 3x+1 Fractal Paper that generalizes the Collatz problem to complex numbers. Includes insights, results and references. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TYG-4C47CDM-5&_coverD

44. Re: [Jprogramming] Collatz Problem Again When you feel the urge to go one step beyond, don't forget to deliver. \/ Randy A MacDonald APL If http://www.mail-archive.com/programming@jsoftware.com/msg03861.html

programming Thread Date Re: [Jprogramming] Collatz Problem again ramacd Sat, 03 Feb 2007 21:07:18 -0800 http://156.34.82.188/ http://www.iisc.ernet.in/academy/resonance/May2005/pdf/May2005Classics.pdf http://www.amazon.com/exec/obidos/ISBN=0387972994/ http://fresh.homeunix.net/~luke/misc/knuth-turingaward.pdf ... http://www.jsoftware.com/forums.htm For information about J forums see http://www.jsoftware.com/forums.htm Re: [Jprogramming] Collatz Problem again Roger Hui Re: [Jprogramming] Collatz Problem again Roger Hui Re: [Jprogramming] Collatz Problem again June Kim RE: [Jprogramming] Collatz Problem again Miller, Raul D Re: [Jprogramming] Collatz Problem again Eric Iverson Re: [Jprogramming] Collatz Problem again ramacd Re: [Jprogramming] Collatz Problem again ramacd Reply via email to

45. The 3n+1 Collatz Problem This problem has a mysterious history and it is not easy to explain the attraction of mathematicians for it. Consider f(n)=n/2 in n is even and http://www.ime.usp.br/~mane/collatz.html

The 3n+1 Collatz Problem This problem has a mysterious history and it is not easy to explain the attraction of mathematicians for it. Consider f(n)=n/2 in n is even and f(n)=(3n+1)/2 if n is odd. It is true that, for all n, the orbit of n by f ends in the cycle (1-2)? The answer for this question is unknown and a lot of problems have borned in the research of this answer (e.g. it is true that the conjecture holds for a set A of positive density? (answer: unknown). Or, if the orbit of n by f is bounded it ends in the cycle (1-2)? (answer: unknown)). We study several aspects of the dynamical system defined by function f and several generalizations of this function. Members Active in this Field (at IME-USP) Manuel Valentim de Pera Garcia - (e-mail mane@ime.usp.br Fabio Armando Tal - (email fabiotal@ime.usp.br Publications A note on the generalized 3n+1 problem - Manuel V. P. Garcia and Fabio A. Tal - Acta Arithmetica XC 3 (1999) pp. 245-250 mane at ime.usp.br Up to research fields in applied math

46. Collatz Problem Goldbach's conjecture Up Unsolved Problems Previous Does there exist a . Collatz Problem. Take any natural number m 0. n= m; repeat if (n is odd) then n=3* n +1; else n http://www.cs.uwaterloo.ca/~alopez-o/math-faq/node61.html

Next: Goldbach's conjecture Up: Unsolved Problems Previous: Does there exist a Collatz Problem Take any natural number m n m repeat n is odd) then n n +1; else n n until ( n The conjecture has been verified for all numbers up to References Unsolved Problems in Number Theory. Richard K Guy. Springer, Problem E16. Elementary Number Theory. Underwood Dudley. 2nd ed. G.T. Leavens and M. Vermeulen 3x+1 search programs Comput. Math. Appl. vol. 24 n. 11 (1992), 79-99. Alex Lopez-Ortiz Mon Feb 23 16:26:48 EST 1998

47. Collatz-Problem – Wikipedia Translate this page Das Collatz-Problem, auch als (3n + 1)-Vermutung bezeichnet, ist ein ungelöstes mathematisches Problem, das 1937 von Lothar Collatz entdeckt wurde. http://de.wikipedia.org/wiki/Collatz-Problem

Collatz-Problem aus Wikipedia, der freien Enzyklopädie Wechseln zu: Navigation Suche Das Collatz-Problem , auch als n -Vermutung bezeichnet, ist ein ungelöstes mathematisches Problem, das 1937 von Lothar Collatz entdeckt wurde. Inhaltsverzeichnis Problemstellung Ursprung und Geschichte Publizierte Quellen Der Collatz-Graph einer Funktion ... Bearbeiten Problemstellung Bei dem Problem geht es um Zahlenfolgen , die nach einem einfachen Bildungsgesetz konstruiert werden: Beginne mit irgendeiner natürlichen Zahl n Ist n gerade, so nimm als nächstes n ist n ungerade, so nimm als nächstes n So erhält man z. B. für die Startzahl n die Folge Anscheinend mündet jede Folge in den Zyklus 4, 2, 1 – egal welche Startzahl man probiert hat. Die Collatz-Vermutung lautet: Jede so konstruierte Zahlenfolge endet im Zyklus 4, 2, 1, egal, mit welcher natürlichen Zahl man beginnt. Trotz zahlreicher Anstrengungen gehört diese Vermutung noch immer zu den ungelösten Problemen der Mathematik . Mehrfach wurden Preise für eine Lösung ausgelobt: 1970 bot H. S. M. Coxeter

48. PlanetMath: Collatz Problem Other names Ulam's Problem, 14-2 Problem, Syracuse problem, Thwaites conjecture, Kakutani's problem, 3n+1 problem http://planetmath.org/encyclopedia/142Problem.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 Collatz problem (Conjecture) CollatzProblem "Collatz problem" is owned by akrowne view preamble get metadata View style: jsMath HTML HTML with images page images TeX source Other names: Ulam's Problem, 1-4-2 Problem, Syracuse problem, Thwaites conjecture, Kakutani's problem, 3n+1 problem Keywords: Collatz, Ulam Attachments: Collatz sequence (Definition) by PrimeFan map (Definition) by PrimeFan Log in to rate this entry. view current ratings There are 4 references to this entry. This is version 16 of Collatz problem , born on 2001-08-19, modified 2003-11-05. Object id is , canonical name is CollatzProblem Accessed 27040 times total. Classification: AMS MSC (Number theory :: Sequences and sets :: Recurrences) Pending Errata and Addenda None.

49. Some Results On The Collatz Problem Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.springerlink.com/index/AW10YK5GR59L0N1E.pdf

50. Collatz Problem@Everything2.com I used to do this in CS/ Math / Physics classes when I was bored. You can easily waste a good twenty minutes per class doing this. But I just found out there's actually a name http://everything2.com/title/Collatz Problem

51. CiteSeerX — The Undecidability Of The Generalized Collatz Problem by SA Kurtz 2006 - Cited by 3 - Related articles http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.84.6811

52. Experiments With The Collatz Problem The Collatz Conjecture (or Collatz problem or 3x+1 problem or hailstorm problem and many more) is an unsolved conjecture in mathematics related to the http://protempore.net/~calvins/misc/collatz/

Exploring the Collatz Problem Home Misc. Introduction Incomplete and in-progress... The Collatz Conjecture (or Collatz problem or 3x+1 problem or hailstorm problem and many more) is an unsolved conjecture in mathematics related to the Collatz function, which is the function that takes a positive integer n to n/2 if n is even or to if n is odd. The conjecture is that for any initial starting number that is a positive integer, iterative application of the function eventually results in For example, if we start with , the successive values are (via (via n/2 . We call the sequence of numbers the Collatz sequence for n = 3 , and the length of this sequence (not including the initial ) is called the Collatz length — in this case, If we start with , then the sequence is . Sometimes the route to is very circuitous. For example, the Collatz sequence for has a length of , and it meanders above at one point. takes steps, and its highest value is more than 27 million! Visualizations of Collatz Length Here are some images that give a sense of how the length of the Collatz sequence varies with n . Click on any image to get a bigger version. Note the spike at n = 27 in the following image: As you can see, there is quite a bit of variety in the lengths, and yet there are regularities too. You can see many examples of two or three consecutive values of

53. The Collatz Problem The Collatz Problem, related functions, data and models http://collatz.freehostia.com/

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54. $Cycles\; Of\; The\; Generalized\; Collatz\; Problem,\; John\; L\; Simons.$ Aug 6, 2009 The generalized Collatz problem (also called the $px+q$ problem with $p$ and $q$ an odd prime) is defined by a sequence of natural numbers http://www.allacademic.com/meta/p377379_index.html

document.write("first");document.write("previous");document.write("top");document.write("next");document.write("last"); FAQ Research All Academic Inc. Citation Cycles of the generalized Collatz Problem, John L Simons. Abstract Word Stems Keywords ... Similar Titles Abstract: The generalized Collatz problem (also called the $px+q$ problem with $p$ and $q$ an odd prime) is defined by a sequence of natural trajectories and exceptionally cyclic behavior. An $m$-cycle of the $px+q$ problem has $m$ local minima $x_i$ and if $(x_i,q)=1$ we call such a cycle primitive. Let $C(m,p,q)$ be the number of number of primitive cycles. We prove for each $m,p,q$ that $C(m,p,q) Convention Need a solution for abstract management? All Academic can help! Contact us today to find out how our system can help your annual meeting. Submission - Custom fields, multiple submission types, tracks, audio visual, multiple upload formats, automatic conversion to pdf. Review - Peer Review, Bulk reviewer assignment, bulk emails, ranking, z-score statistics, and multiple worksheets! Reports - Many standard and custom reports generated while you wait. Print programs with participant indexes, event grids, and more!

55. Halfbakery: Solve The Collatz Problem With Wave Mechanics Ok, so I noticed a similarity between the waves on the surface of my dogs water bowl as I filled it as cataloged in the original linked idea. http://www.halfbakery.com/idea/Solve_20the_20Collatz_20Problem_20with_20Wave_20M

h a l f b a k e r y Not quite a think tank. More of a think punch bowl. idea: add search annotate link , view, overview recent by name best ... random meta: news help about links ... report a problem account: browse anonymously, or get an account and write. user: pass: register polyhedral sections Privatise the numbers system Quicker Counting ... mathematics Solve the Collatz Problem with Wave Mechanics Who knows, Maybe? [vote for against Ok, so I noticed a similarity between the waves on the surface of my dogs water bowl as I filled it as cataloged in the original linked idea. I realized as I was filling the bowl again the other day that its not the fluid dynamics which produce those patterns, but the wave interference. Perhaps the wave mechanics could be applied to the stated problem to find a solution. I'm expecting bones on this one, but I'm not an advanced enough mathematician to know how to apply this similarity, but perhaps someone out there is. bleh , Aug 30 2007 A plot of stopping times for the Collatz Problem http://www.cs.cmu.e...cs/Collatz-stop.jpg notice the similarity to wave interference patterns [ bleh , Aug 30 2007] Similar wave interference http://www.colorado...es/interference.gif

56. Collatz Conjecture (3x+1 Problem) « SendDerek’s Blog Oct 1, 2007 The Collatz Conjecture, also known as the 3x+1 problem, is simple by definition, but upon further research it becomes a very complex theory. http://sendderek.wordpress.com/2007/10/01/collatz-conjecture-3x1-problem/

57. The Dual Collatz Problem - Microsoft Academic Search Authors John L Simons. Citations 0 The generalized Collatz problem is defined by a sequence of natural numbers, generated conditionally by xn+1 = 12xn if xn is even and by xn+1 http://academic.research.microsoft.com/Paper/4910466.aspx

var SiteRoot = 'http://academic.research.microsoft.com'; SHARE Author Conference Journal Year Look for results that meet for the following criteria: since equal to before Publication The dual Collatz problem Edit The dual Collatz problem John L Simons View or Download The following links allow you to view and download full papers. These links are maintained by other sources not affiliated with Microsoft Academic Search. www.rug.nl Reference Formes lin'eaires en deux logarithmes et d'eterminants d''interpolation Citations: 53 ... Yu. Nesterenko Published in 1995. On the (non)-existence of m-cycles for generalized Syracuse sequences Citations: 1 John L Simons The generalized 3x + 1 mapping ... K. R. Matthews Published in 2000. Theoretical and computational bounds for m-cycles of the 3n + 1 problem Citations: 7 John Simons Published in 2004. Microsoft Research Asia Help Feedback Specific Terms ... Privacy Statement Share this on Contribute to Academic

58. Recent Developments In Solving The Generalized Collatz Problem File Format PDF/Adobe Acrobat Quick View http://www.rug.nl/staff/j.l.simons/recdevslides.pdf

59. Anarchy Golf - Collatz Problem Language is selected by the extension of the file. See the list of supported languages to know the extension of your language. Problem. Collatz Problem http://golf.shinh.org/p.rb?Collatz Problem

60. Bookmarks 3xp1 problem syracuse collatz This paper reports on some functional equations, which arise in connection with the famous 3n+1 Collatz problem. In particular, it reports on two analytic http://jean-paul.davalan.pagesperso-orange.fr/liens/liens_syracuse.html

Accueil Plan du site Bloc notes Jeux ... .net Votre portail e-Learning Culture MATH ENSup. et Minist. EN On-Line Encyclopedia of Integer Sequences Séminaire MaMuX Mathématiques, musique et relations avec d'autres disciplines L'INVERSEUR de Plouffe Inscrire les services publics dans la constitution Plus de 210000 signatures (27 mai) _stats("Jeux","Liens","syracuse") 3x+1, Collatz, Syracuse Problem SOMMAIRE Pages Web Maths Demos ... Liens 3x+1 problem applet Cliquer l'intérieur du cadre de l'applet pour changer d'élément. Voir aussi la Page d'origine de l'applet Paragraphe DEMOS PAGES WEB The 3x+1 problem and its generalizations Computational verification of the 3x+1 conjecture Tomás Oliveira e Silva Computational projects On the 3x+1 problem By Eric Roosendaal Glide Records Table. Delay Records Table residues page completeness page ... Super-Index of The 3x+1 Problem and its generalizations This index was automatically generated using a new tagging program written by Simon Plouffe at the CECM, Simon Fraser University. The Collatz (3x + 1) Problem Ilan Vardi Collatz Problem The 3x+1 Problem The Collatz Problem (3x+1) Evans A Criswell On the 3x+1 problem By Eric Roosendaal Unsolved problems Ken Monks's FSRP Page Keith Matthews' 3x+1 page Homepage of Keith Matthews

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