1. Collatz Problem A problem first posed by the German mathematician Lothar Collatz (1910–1990) in 1937, that is also known variously as the 3n + 1 problem, Kakutani s problem http://www.daviddarling.info/encyclopedia/C/Collatz_problem.html

2. Collatz Problem – Rechenkraft The Collatz Problem and Analogues Bart Snapp and Matt Tracy Department of Mathematics and Statistics Coastal Carolina University Conway, SC 295286054 http://www.rechenkraft.net/wiki/index.php?title=Collatz_Problem

3. Number Theory > Open Problems > Collatz Problem International Conference on the Collatz Problem and Related Topics August 56, 1999 Katholische Universitat Eichstatt, GERMANY This conference is intended http://www.einet.net/directory/962657/Collatz_Problem.htm

4. Wolfram|Alpha: Collatz Problem : Statement, Proof, Status, Awards, ... Complete information and computations for Collatz problem basic properties, history, http://www.wolframalpha.com/entities/famous_math_problems/collatz_problem/wr/u9/

document.write(''); Home Examples About FAQS ... Media Resources A Wolfram Web Resource Wolfram Research Wolfram Mathematica Wolfram Demonstrations ... Wolfram Tones Enter something to compute or figure out Calculate Collatz problem: statement, proof, status, awards, formulation, evidence, ... Input interpretation: Statement: Alternate names: History: More Current evidence: Associated prizes: Computed by Wolfram Mathematica Download as: PDF Live Mathematica A Wolfram Research Company Participate Entity Index Collatz problem: statement, proof, status, awards, formulation, evidence, ... Input: collatz problem Collatz problem Statement Alternate names 3n+1 problem 3x+1 mapping Hasse(close curly quote)s algorithm Kakutani(close curly quote)s problem Syracuse algorithm Syracuse problem Thwaite(close curly quote)s conjecture Ulam(close curly quote)s problem History Current evidence Associated prizes

5. Collatz Problem - AoPSWiki Define the following function on The Collatz conjecture says that, for any positive integer, the sequence contains 1. This conjecture is still open. http://www.artofproblemsolving.com/Wiki/index.php/Collatz_Problem

Art of Problem Solving LOGIN/REGISTER Home Bookstore AoPS Curriculum ... Articles You are here: Resources AoPSWiki Search Wiki Page Tools Page Discussion View source History Personal tools Talk for this IP Log in Navigation Main Page Community portal Current events Recent changes ... Help Toolbox What links here Related changes Special pages Printable version ... Permanent link Collatz Problem From AoPSWiki Define the following function on The Collatz conjecture says that, for any positive integer , the sequence contains 1. This conjecture is still open. Some people have described it as the easiest unsolved problem in mathematics. This article has been found by the AoPSWiki Editors not to be written in wiki style. Help us out by putting this page in wiki style and removing this message. Retrieved from " http://www.artofproblemsolving.com/Wiki/index.php/Collatz_Problem Categories Articles that need to be wikified Number theory ... Conjectures This page was last modified on 15 February 2009, at 15:22. This page has been accessed 2,189 times. About AoPSWiki

6. Java Programming - Collatz Problem hi i have to trying to work onsolving collatz problem http//en.wikipedia.org/wiki/collatz_problem the code i want to make is to find the number less than 1 million which will form http://forums.sun.com/thread.jspa?threadID=5129481&tstart=49740

7. Java Programming - Collatz Problem 3 posts 3 authors - Last post Jan 27, 2007hi i have to trying to work onsolving collatz problem http//en.wikipedia.org/ wiki/collatz_problem the code i want to make is to find the http://forums.sun.com/thread.jspa?threadID=5129481

8. Wapedia - Wiki: Collatz Conjecture In 2006, researchers Kurtz and Simon, building on earlier work by J.H. Conway in the 1970s, proved that a natural generalization of the Collatz problem is http://wapedia.mobi/en/Collatz_problem

Wiki: Collatz conjecture The Collatz conjecture is an unsolved conjecture in mathematics named after Lothar Collatz , who first proposed it in 1937. The conjecture is also known as the n + 1 conjecture , the Ulam conjecture (after Stanisław Ulam (after Shizuo Kakutani ), the Thwaites conjecture (after Sir Bryan Thwaites), (after Helmut Hasse ), or the Syracuse problem the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers or as wondrous numbers Directed graph showing the orbits of small numbers under the Collatz map. The Collatz conjecture is equivalent to the statement that all paths eventually lead to 1. Take any natural number n . If n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to obtain 3 n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. It has been called "Half Or Triple Plus One", sometimes called HOTPO The property has also been called oneness Directed graph showing the orbits of the first 1000 numbers.

9. Collatz Problem -- From Wolfram MathWorld Oct 11, 2010 From Eric Weissten s World of Mathematics. Article with references and links. http://mathworld.wolfram.com/CollatzProblem.html

10. 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

11. Collatz Conjecture - Wikipedia, The Free Encyclopedia Bruschi, Mario, A generalization of the Collatz problem and conjecture, 29 October 2008, Number Theory category in arXiv preprint collection at Cornell University Library. http://en.wikipedia.org/wiki/Collatz_conjecture

Collatz conjecture From Wikipedia, the free encyclopedia Jump to: navigation search Directed graph showing the orbits of small numbers under the Collatz map. The Collatz conjecture is equivalent to the statement that all paths eventually lead to 1. Directed graph showing the orbits of the first 1000 numbers. The Collatz conjecture is an unsolved conjecture in mathematics named after Lothar Collatz , who first proposed it in 1937. The conjecture is also known as the n + 1 conjecture , the Ulam conjecture (after Stanisław Ulam Kakutani's problem (after Shizuo Kakutani ), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse ), or the Syracuse problem the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers or as wondrous numbers Take any natural number n . If n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to obtain 3 n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. It has been called "Half Or Triple Plus One", sometimes called HOTPO The property has also been called oneness Paul Erdős said about the Collatz conjecture: "Mathematics is not yet ready for such problems." He offered $500 for its solution.

12. 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/CollatzProblem.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 27039 times total. Classification: AMS MSC (Number theory :: Sequences and sets :: Recurrences) Pending Errata and Addenda None.

13. 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:

14. The Collatz Problem (3x+1) The Collatz Problem (3x+1) I was introduced to the Collatz problem back in 1990 by Dr. Ashok T. Amin here in the Computer Science Department at the University of Alabama in http://hsvmovies.com/static_subpages/personal_orig/math/collatz.html

The Collatz Problem (3x+1) I was introduced to the Collatz problem back in 1990 by Dr. Ashok T. Amin here in the Computer Science Department at the University of Alabama in Huntsville. Dr. Niall Graham, also here in the department, has recently revived my interest in it. The problem deals with sequences of integers generated as follows: Start with a positive integer x > 0. Repeat the following steps: If the last integer in the sequence is 1, stop. The sequence is complete. If the last integer in the sequence is even, divide it by two to get the next integer in the sequence. If the last integer in the sequence is odd, multiply it by three and add one to get the next integer in the sequence. The problem is very simple to state, and the actions are very simple to perform, but the question is, given any starting integer x > 0, will the sequence generated end with the integer 1 in a finite number of steps? Here are the sequences generated for the first few integers: Here is, perhaps, a neater way of showing it: (under construction) As you can see, they all end up at 1. It is interesting to turn this problem around and look at it in reverse, starting with 1 and going in reverse to produce sequences. The reverse of the procedure above is the following:

15. 3x+1 Conjecture Verification Results Results up to 2^60 by Tom s Oliveira e Silva. http://www.ieeta.pt/~tos/3x 1.html

16. Debugging Example--The Collatz Problem :: Editing And Debugging M-Files (Desktop Debugging ExampleThe Collatz Problem. The example debugging session requires you to create two Mfiles, collatz.m and collatzplot.m, that produce data for the Collatz problem. http://matlab.izmiran.ru/help/techdoc/matlab_env/edit_d27.html

Desktop Tools and Development Environment Debugging ExampleThe Collatz Problem The example debugging session requires you to create two M-files, collatz.m and collatzplot.m , that produce data for the Collatz problem. For any given positive integer, n , the Collatz function produces a sequence of numbers that always resolves to 1. If n is even, divide it by 2 to get the next integer in the sequence. If n is odd, multiply it by 3 and add 1 to get the next integer in the sequence. Repeat the steps until the next integer is 1. The number of integers in the sequence varies, depending on the starting value, n The Collatz problem is to prove that the Collatz function will resolve to 1 for all positive integers. The M-files for this example are useful for studying the Collatz problem. The file collatz.m generates the sequence of integers for any given n . The file collatzplot.m calculates the number of integers in the sequence for all integers from 1 through m , and plots the results. The plot shows patterns that can be further studied. Following are the results when n is 1, 2, or 3.

17. The 3x+1 Problem And Its Generalizations Abstract (taken from the Introduction) The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns http://www.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

18. 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

19. Project Euler Question 14 (Collatz Problem) - Stack Overflow The following iterative sequence is defined for the set of positive integers n n/2 (n is even) n - 3n + 1 (n is odd) Using the rule above and starting with 13, we generate http://stackoverflow.com/questions/2643260/project-euler-question-14-collatz-pro

20. Collatz Problem - Application Center Collatz Problem The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of http://www.maplesoft.com/applications/view.aspx?SID=1665

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