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Collatz Problem:     more detail

Conjectures: P Versus Np Problem, Poincaré Conjecture, Collatz Conjecture, Hilbert's Problems, Catalan's Conjecture, Sierpinski Number Optimization Problems (Applied Mathematical Sciences) (Volume 0) by L. Collatz, W. Wetterling, 1975-07-23 Numerical Treatment of Free Boundary Value Problems: WORKSHOP NUMER.TREATMEnt Free Bound.Value Probl.Oberwolfach (International Series of Numerical Mathematics) by ALBRECHT, COLLATZ, et all 1982-06-14 Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations: CONFERENCE AT THE OBERwolfach Math.Res.Inst.,19.-25.11.1978 (International Series of Numerical Mathematics) by ALBRECHT, COLLATZ, et all 1979-01-01 Numerical Treatment of Eigenvalue Problems, Vol. 3 (International Series of Numerical Mathematics) by ALBRECHT, COLLATZ, et all 1984-05-31 NUMERICAL TREATMENT OF FREE BOUNDARY VALUE PROBLEMS by J.; Collatz, L.; et al; editors Albrecht, 1982 Numerical Treatment of Eigenvalue Problems: Workshop in Oberwafach, February 25-March 3, 1990/Numerische Behandlung Von Eigenwertaufgaben : Tagung (International Series of Numerical Mathematics) by J. Albrecht, Lothar Collatz, et all 1991-05 Numerical Treatment of Eigenvalue Problems (International Series of Numerical Mathematics) OPTIMIZATION PROBLEMS.Translated by P.Wadsack.*(Applied Mathematical Sciences,17) by L./Wetterling,W. Collatz, 1975-01-01 The Dynamical System Generated by the 3n+1 Function (Lecture Notes in Mathematics) by Günther J. Wirsching, 1998-03-20

lists with details

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
    
    Extractions: new recent math what is this? what is this? 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] From: Mario Bruschi [ view email

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/
    
    Extractions: 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.

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
    
    Extractions: "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

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
    
    Extractions: Printer-friendly page - just print Andrew Shapira February 15, 1998 Includes minor subsequent revisions such as web link updates. 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
    
    Extractions: (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 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.

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

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
    
    Extractions: (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 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.

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
    
    Extractions: 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
    
    Extractions: new recent what is this? 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] From: Craig Alan Feinstein [ view email

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