21. [0810.5169] A Generalization Of The Collatz Problem And Conjecture Abstract We introduce an infinite set of integer mappings that generalize the wellknown Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings http://arxiv.org/abs/0810.5169

arXiv.org math Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PDF PostScript Other formats Current browse context: math.NT new recent Change to browse by: math NASA ADS 1 blog link what is this? Bookmark what is this? Mathematics > Number Theory Title: A generalization of the Collatz problem and conjecture Authors: M. Bruschi (Submitted on 29 Oct 2008) Abstract: We introduce an infinite set of integer mappings that generalize the well-known Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings feature the remarkable property of the Collatz conjecture, namely that they converge to unity irrespective of which positive integer is chosen initially. Subjects: Number Theory (math.NT) Cite as: arXiv:0810.5169v1 [math.NT] Submission history From: Mario Bruschi [ view email Wed, 29 Oct 2008 00:01:45 GMT (5kb) Which authors of this paper are endorsers? Link back to: arXiv form interface contact

22. On The 3x + 1 Problem This latter formula usually gives the sequence its name, the 3x + 1 problem, sometimes also referred to as the Collatz problem, the Syracuse problem or some http://www.ericr.nl/wondrous/

On the 3x + 1 problem By SUMMARY: The so-called 3x+1 problem is to prove that all 3x+1 sequences eventually converge. The sequences themselves however and their lengths display some interesting properties and raise unanswered questions. These pages supply numerical data and propose some conjectures on this innocent looking problem. This page contains the following sections: Definition of the problem The Glide Delay and Residue Completeness and Gamma ... distributed search for new class records! progress of the distributed search project Site Map Latest Path Record news: The last new Path Record confirmed was in November 2009. This record occurs at , (or ) and it reaches a maximum of In October 2010 the mark of 28 . 2 was reached. A complete list of Path records is on the Path records page This page was last modified on October 24, 2010 Introduction and definitions For any positive integer N a sequence S i can be defined by putting S = N and for all S i = S i-1 if S i-1 is even S i = S i-1 if S i-1 is odd This latter formula usually gives the sequence its name, the

23. Collatz Problem As A Cellular Automaton - Wolfram Demonstrations Project The Collatz problem asks whether 1 is reached. In all known cases, the sequence always reaches 1, but no proof is known that this is always true. http://demonstrations.wolfram.com/CollatzProblemAsACellularAutomaton/

24. Notes On Lookup – Another Sieve For The Collatz Problem « Learning And Unlear This post is a followup on an earlier post in which I introduced the Collatz Problem and designed a sieve that systematically builds solutions and is very efficient in the work it http://unlearningmath.com/2009/02/27/notes-on-lookup-another-sieve-for-the-colla

25. 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.

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

27. Onezero » An Image From The Collatz Problem By Andrew Shapira. The intensity of a point denotes the time taken to terminate. http://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:

28. Collatz Problem Definition of Collatz problem, possibly with links to more information and implementations. http://www.itl.nist.gov/div897/sqg/dads/HTML/CollatzProblem.html

Collatz problem (classic problem) Definition: Note: First posed by the German mathematician Lothar Collatz in 1937. Began appearing in print in the 1950's. The sequence of integers generated are called "hailstones", since they may rise and fall but (presumably) always fall to the ground (the integer 1). Also called the Syracuse problem and many other names. Author: PEB More information MathWorld's Collatz problem entry. Jeff Lagarias The 3x+1 Problem and its Generalizations , American Mathematical Monthly, 92:3-23, 1985. This paper gives the history of the problem, including various names, many references, and a survey of what is (was) known. The web version also has links to related sites. Go to the Dictionary of Algorithms and Data Structures home page. If you have suggestions, corrections, or comments, please get in touch with Paul E. Black Entry modified 14 August 2008. HTML page formatted Mon Sep 27 10:31:22 2010. Cite this as: Paul E. Black, "Collatz problem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology

29. Collatz Problem | Plus.maths.org Many of our readers have asked for more information about the hailstrone sequence problem from the last issue. http://plus.maths.org/content/taxonomy/term/1195

Skip to Navigation about Plus support Plus Plus sponsors ... subscribe to Plus Search this site: collatz problem More hailstones... Many of our readers have asked for more information about the hailstrone sequence problem from the last issue. Read more... Home Articles Careers ... view Login to comment or download PDFs Username: Password: Create new account Request new password

30. Collatz Problem collatz problem Number Theory discussion in mathworld, they say that conway proved that Collatztype problems can be formally undecidable. http://www.physicsforums.com/showthread.php?p=418685

31. YouTube - Collatz Problem As A Cellular Automaton http//demonstrations.wolfram.com/Col The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. Start with an http://www.youtube.com/watch?v=d2UcGuCXkhw

32. Collatz Problem Definition of Collatz problem, possibly with links to more information and implementations. http://xw2k.nist.gov/dads/HTML/CollatzProblem.html

Collatz problem (classic problem) Definition: Note: First posed by the German mathematician Lothar Collatz in 1937. Began appearing in print in the 1950's. The sequence of integers generated are called "hailstones", since they may rise and fall but (presumably) always fall to the ground (the integer 1). Also called the Syracuse problem and many other names. Author: PEB More information MathWorld's Collatz problem entry. Jeff Lagarias The 3x+1 Problem and its Generalizations , American Mathematical Monthly, 92:3-23, 1985. This paper gives the history of the problem, including various names, many references, and a survey of what is (was) known. The web version also has links to related sites. Go to the Dictionary of Algorithms and Data Structures home page. If you have suggestions, corrections, or comments, please get in touch with Paul E. Black Entry modified 14 August 2008. HTML page formatted Mon Sep 27 10:31:22 2010. Cite this as: Paul E. Black, "Collatz problem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology

33. Collatz Problem Complete learning touching collatz problem. You are able dig up some information touching news25 too . http://members.multimania.nl/uucikrq/collatz-problem.html

34. 3n+1 Sequences An online calculator by Alfred Wassermann. http://did.mat.uni-bayreuth.de/personen/wassermann/fun/3np1_e.html

Universität Bayreuth - Lehrstuhl für Mathematik und ihre Didaktik - Universitätsstraße 30 - 95447 Bayreuth - Fon: +49 921 55 32 66 - Fax: +49 921 55 21 61 Lehrstuhl für Mathematik und ihre Didaktik Startseite Forschungsprojekte Kooperationen Studium ... Zusatzqualifikation Multimediakompetenz Experiments with the 3n+1 sequence Links Go Key Resource Number Theory Topic Deutsche Version Starting with the natural number n , a sequence of numbers will be computed: If the number is even, it will be divided by 2. If the number is odd, it will be multiplied by 3 and then 1 will be added. Example: Input of the number 17. 17 is odd, it becomes: 3 * 17 + 1 = 52 52 is even, it becomes: 52 / 2 = 26 26 is even, it becomes 13 13 is odd, it becomes 40 40 is even, it becomes 20 20 is even, it becomes 10 10 is even, it becomes 5 5 is odd, it becomes 16 16 is even, it becomes 8 8 is even, it becomes 4 4 is even, it becomes 2 2 is even, it becomes 1 If the sequence reaches 1, then it is catched in an infinite loop: 1 becomes 4, 4 becomes 2 and 2 finally becomes 1 again. Until now all numbers which were tested eventually ended up with 1. But there is no proof known that this will be be the case for all natural numbers. Here you can test your own numbers:

35. The Undecidability Of The Generalized Collatz Problem File Format PDF/Adobe Acrobat Quick View http://people.cs.uchicago.edu/~simon/RES/collatz.pdf

36. Collatz Conjecture Calculation Center Calculation programs (JavaScript) reflecting a new mathematical approach of the Collatz problem. In addition some general math tools and some math tools for the odd perfect http://members.chello.nl/k.ijntema/

37. The 3x+1 Problem Annotated Bibliography By Jeffrey Lagarias, 1997. http://www.math.lsa.umich.edu/~lagarias/3x 1.html

38. [math/0312309] The Collatz 3n+1 Conjecture Is Unprovable A paper by Craig Alan Feinstein arguing that the Collatz Conjecture cannot be formally proved. http://arxiv.org/abs/math.GM/0312309

arXiv.org math Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PDF PostScript Other formats Current browse context: math new recent NASA ADS Bookmark what is this? Mathematics > General Mathematics Title: The Collatz 3n+1 Conjecture is Unprovable Authors: Craig Alan Feinstein (Submitted on 16 Dec 2003 ( ), last revised 4 May 2006 (this version, v16)) Abstract: In this paper, we show that any proof of the Collatz 3n+1 Conjecture must have an infinite number of lines; therefore, no formal proof is possible. We also give an informal argument that the Riemann Hypothesis is also unprovable. Comments: 3 pages. Modified the proof of Theorem 2 Subjects: General Mathematics (math.GM) MSC classes: Cite as: arXiv:math/0312309v16 [math.GM] Submission history From: Craig Alan Feinstein [ view email Tue, 16 Dec 2003 16:45:16 GMT (5kb) Fri, 9 Jan 2004 18:27:15 GMT (5kb) Wed, 14 Jan 2004 16:40:51 GMT (5kb) Thu, 15 Jan 2004 20:59:17 GMT (5kb) Tue, 20 Jan 2004 18:36:16 GMT (5kb) Fri, 23 Jan 2004 00:36:42 GMT (5kb)

39. The 3x+1 Problem And Its Generalizations An activated-text survey article by Jeff Lagarias. http://oldweb.cecm.sfu.ca/organics/papers/lagarias/

The 3x+1 problem and its generalizations*** Jeff Lagarias Murray Hill, New Jersey Math activated text Other available formats Related links ... Author biography Abstract: (taken from the Introduction) The problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm , and Ulam's problem , concerns the behavior of the iterates of the function which takes odd integers n to and even integers n to n/2 . The Conjecture asserts that, starting from any positive integer n , repeated iteration of this function eventually produces the value The Conjecture is simple to state and apparently intractably hard to solve. It shares these properties with other iteration problems, for example that of aliquot sequences and with celebrated Diophantine equations such as Fermat's last theorem. Paul Erdos commented concerning the intractability of the problem: "Mathematics is not yet ready for such problems." Despite this doleful pronouncement, study of the problem has not been without reward. It has interesting connections with the Diophantine approximation of the binary logarithm of and the distribution mod 1 of the sequence , with questions of ergodic theory on the -adic integers, and with computability theory - a generalization of the

40. The Collatz Problem And Analogues File Format PDF/Adobe Acrobat Quick View http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Snapp/snapp.pdf

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