1. Collatz Problem -- From MathWorld From Eric Weissten s World of Mathematics. Article with references and links. http://mathworld.wolfram.com/CollatzProblem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Sequences Foundations of Mathematics ... Martinez Collatz Problem A problem posed by L. Collatz in 1937, also called the mapping, conjecture . Let be an integer . Then the Collatz problem asks if iterating always returns to 1 for positive . The members of the sequence produced by the Collatz are sometimes known as hailstone numbers . Conway proved that the original Collatz problem has no nontrivial cycles of length . Lagarias (1985) showed that there are no nontrivial cycles with length . Conway (1972) also proved that Collatz-type problems can be formally undecidable The following table gives the sequences obtained for the first few starting values (Sloane's The numbers of steps required for the algorithm to reach 1 for , 2, ... are 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, ... (Sloane's ; illustrated above). Of these, the numbers of tripling steps are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane's

2. Collatz Problem Image By Andrew Shapira. The intensity of a point denotes the time taken to terminate. http://www.onezero.org/collatz.html

An Image From the Collatz Problem By Andrew Shapira February 15, 1998 (Minor revisions such as web link updates were made subsequently.) 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. It has been conjectured that all integers eventually yield a 1. The ``Collatz problem'', also known as the ``3x+1'' problem, is to determine whether the conjecture is true. The conjecture has been verified by computer up to . (See the table of contents at the sci.math FAQ and follow the link to ``Unsolved Problems.'') 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:

3. The 3x+1 Problem And Its Generalizations The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. Unsolved Problems -- From MathWorld The collatz problem. 8. Proof that the 196algorithm does not terminate when applied to the number 196. 9. Proof that 10 is a solitary number. http://mathworld.wolfram.com/UnsolvedProblems.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Mathematical Problems Unsolved Problems Unsolved Problems There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include 1. The Goldbach conjecture 2. The Riemann hypothesis 3. The 4. The conjecture that there exists a Hadamard matrix for every positive multiple of 4. 5. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes 6. Determination of whether NP-problems are actually P-problems 7. The Collatz problem 8. Proof that the 196-algorithm does not terminate when applied to the number 196. 9. Proof that 10 is a solitary number 10. Finding a formula for the probability that two elements chosen at random generate the symmetric group 11. Solving the

5. 3x+1 Conjecture Verification Results References 1 Jeffrey C. Lagarias, The 3x+1 problem and its generalizations, The American Mathematical Monthly, vol. 92, no. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. 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&

7. On The 3x + 1 Problem usually gives the sequence its name, the 3x + 1 problem, sometimes also referred to as the collatz problem, the Syracuse problem or some http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. International Conference On The Collatz Problem Katholische Universität Eichstätt, Germany; 56 August 1999. Online 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

9. The Collatz Problem Experimental Data And Model Properties and predictive models for the collatz problem and relative functions. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Proceedings Of 3x+1 Conference Proceedings of the International Conference on the collatz problem and Related Topics. Some of the participants of the conference offer either a paper or http://www.math.grin.edu/~chamberl/conference/proceedings.html

Proceedings of the International Conference on the Collatz Problem and Related Topics Some of the participants of the conference offer either a paper or abstract relating to their talk at the conference. The (3x+1)/2 Problem and its generalisation: a stochastic approach (postscript) M.R. Feix and J.L. Rouet Some Properties of the 3n+1 Function Using Number Representation (postscript) R. Banerji The 3x+1 Problem and Directed Graphs (postscript) P. Andaloro A Dynamical Systems Approach to the 3x+1 Problem (postscript) M. Chamberland The 3n+1-Problem and Holomorphic Dynamics (postscript) D. Schleicher A Category of Topological Spaces Encoding Acyclic Set-Theoretic Dynamics (postscript) (pdf) K. Monks Some Results on the Collatz Problem (postscript) M. Kudlek Left to right: K. M. Monks, K. G. Monks, P. Andaloro, G. Wirsching, M. Kudlek, R. Banerji, J. Lagarias, D. Schleicher, M. Chamberland, J. L. Rouet, E. Roosendaal, U. Fitze, M. R. Feix, E. G. Belaga

11. Mathematics Disclaimer 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/

Mathematics All of UQ Select a quick link: Other Mathematics Links MathSciNet School of Physical Sciences Faculty of EPSA About UQ my.UQ mySI-net Programs and Courses UQ News UQ Experts UQ Images Organisational Units Staff Web Page You have requested the document http://www.maths.uq.edu.au/~krm/ . This is a staff web area hosted on a University of Queensland web server. Please be advised that the web pages within this area are NOT officially endorsed by The University of Queensland. The University accepts no responsibility or liability for the contents of this area. This message has been displayed in accordance with the University's Internet Code of Practice , which forms a part of the Please note that you will need to enable cookies in your browser in order to proceed. to continue, or to the Mathematics home page. privacy feedback © 2004 The University of Queensland, Brisbane, Australia ABN 63 942 912 684 CRICOS Provider No: Authorised by: Head of School Maintained by: webmaster@sps.uq.edu.au Last Updated - Today 09:31pm

12. Papers On The 3x + 1 / 3n + 1 Problem, Fermat's Last Theorem, And A paper by Peter Schorer describing a new approach to the 3x + 1 Problem, an approach based on two remarkably simple structures that underlie the 3x http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. Collatz Problem collatz problem. collatz problem. Take any natural number m 0. n=m; repeat if (n is odd) then n=3*n+1; else n=n/2; until (n1) http://db.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

14. Collatz Problem From MathWorld A problem posed by L. Collatz in 1937, also called the 3x+1 mapping, 3n+1 problem, Hasse's algorithm, Kakutani's problem, Syracuse algorithm http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. Unsolved Problems collatz problem. Take any natural number m 0. n = m; repeat if (n is odd) then n = 3*n + 1; else n = n/2; until (n1). Conjecture 1. http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node30.html

Next: Mathematical Games Up: Famous Problems in Mathematics Previous: Which are the 23 Unsolved Problems Does there exist a number that is perfect and odd? A given number is perfect if it is equal to the sum of all its proper divisors. This question was first posed by Euclid in ancient Greece. This question is still open. Euler proved that if N is an odd perfect number, then in the prime power decomposition of N , exactly one exponent is congruent to 1 mod 4 and all the other exponents are even. Furthermore, the prime occurring to an odd power must itself be congruent to 1 mod 4. A sketch of the proof appears in Exercise 87, page 203 of Underwood Dudley's Elementary Number Theory. It has been shown that there are no odd perfect numbers Collatz Problem Take any natural number n : = m; repeat n is odd) then n : = 3*n + 1 ; else n : = n/2 until ( n==1 Conjecture 1. For all positive integers m, the program above terminates. The conjecture has been verified for all numbers up to References Unsolved Problems in Number Theory.

16. Voila - Mon Site Properties and predictive models for the collatz problem and relative functions. http://site.voila.fr/Collatz_Problem

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17. The Proof Of The Collatz Problem The third step is practiced the collatz problem Then By My Formula and we omit it because a n dosen't equal zero http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. Finding Errors In M-Files :: Editing And Debugging M-Files (Desktop Tools And De The collatz problem is to prove that the Collatz function will resolve to 1 The Mfiles for this example are useful for studying the collatz problem. http://www.mathworks.com/access/helpdesk/help/techdoc/matlab_env/edit_d27.html

Desktop Tools and Development Environment Finding Err ors in M-Files D e bugging is the process by which you isolate and fix problems with your code. Debugging helps to correct two kinds of errors: Syntax errorsFor example, misspelling a function name or omitting a parenthesis. Run-time errorsThese errors are usually algorithmic in nature. For example, you might modify the wrong variable or code a calculation incorrectly. Run-time errors are usually apparent when an M-file produces unexpected results. Run-time errors are difficult to track down because the function's local workspace is lost when the error forces a return to the MATLAB base workspace. In addition to finding and fixing problems with your M-files, you might want to improve the performance and make other enhancements using MATLAB tools. Use the following techniques to isolate the causes of errors and improve your M-files. Technique or Tool Description For More Information Syntax highlighting Syntax highlighting helps you identify syntax errors in an M-file before you run the file. Syntax Highlighting Error messages When you run an M-file with a syntax error, MATLAB will most likely detect it and display an error message in the Command Window describing the error and showing its line number in the M-file. Click the underlined portion of the error message, or position the cursor within the message and press

19. International Conference On The Collatz Problem International Conference on the collatz problem and Related Topics http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

20. Editing And Debugging M-Files (Development Environment) The collatz problem is to prove that the Collatz function will resolve to 1 for function collatzplot(n) % Plot length of sequence for collatz problem http://www.mathworks.com/access/helpdesk_r13/help/techdoc/matlab_env/edit_d21.ht

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 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 any given integer and plots the results. The plot shows patterns that can be further studied. Following are the results when n is 1, 2, or 3.

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