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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

1. Collatz Problem -- From MathWorld
   collatz problem. Let be an integer. Then the collatz problem asks ifiterating, (1). always returns to 1 for positive . The members  
   http://mathworld.wolfram.com/CollatzProblem.html
   
   Extractions: 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

2. Collatz Problem Image
   An Image From the collatz problem. By Andrew Shapira. February 15, 1998.(Minor We can do the same thing for the collatz problem. Given  
   http://www.onezero.org/collatz.html
   
   Extractions: 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. Unsolved Problems -- From MathWorld
   number of twin primes). 6. Determination of whether NPproblems areactually P-problems. 7. The collatz problem. 8. Proof that the  
   http://mathworld.wolfram.com/UnsolvedProblems.html
   
   Extractions: 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

4. International Conference On The Collatz Problem
   International Conference on the collatz problem and Related TopicsAugust 56, 1999 Katholische Universität Eichstätt, GERMANY.  
   http://www.math.grin.edu/~chamberl/conf.html

5. The 3x+1 Problem
   The 3x+1 Problem. The 3x+1 problem, also known as the collatz problem, theSyracuse problem, Kakutani s problem, Hasse s algorithm, and Ulam s problem  
   http://www.math.grin.edu/~chamberl/3x.html
   
   Extractions: "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

6. 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/
   
   Extractions: BCMath : Online BC number theory programs Publications Unpublished Articles A survey on the generalized 3x+1 (Collatz) mapping ( pdf 206K LLL page Some past number theory lecture courses: Elementary Linear Algebra , An online textbook by Keith Matthews Some past linear algebra lecture courses weblogs MathSciNet (for subscribers only) Google An early history of the UQ Mathematics Department University of Queensland Maps UQ library catalogue ... Html Information (How to write html, forms, tables, validate html pages) Australian and other links Public Transport timetables Phone: my office, Room 449, Priestley Bldg.

7. AlDamen
   Chemistry student at Jerash University with interests in number theory and the collatz problem.  
   http://www.angelfire.com/de2/abbas

8. On The 3x + 1 Problem
   1 if S i1 is odd This latter formula usually gives the sequence its name, the 3x+ 1 problem, sometimes also referred to as the collatz problem, the Syracuse  
   http://personal.computrain.nl/eric/wondrous/
   
   Extractions: By Eric Roosendaal 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.

9. The Collatz Problem, Data And Models
   Properties and predictive models for the collatz problem and relative functions.  
   http://site.voila.fr/Collatz_Problem

10. The 3x+1 Problem And Its Generalizations
    The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Kakutani sproblem, Hasse s algorithm, and Ulam s problem, concerns the behavior of  
    http://www.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

11. PlanetMath: Collatz Problem
    collatz problem, (Conjecture). We define the function (where excludeszero) such that. This is generally called the collatz problem.  
    http://planetmath.org/encyclopedia/142Problem.html
    
    Extractions: Collatz problem (Conjecture) We define the function (where excludes zero) such that Then let the sequence be defined as , with an arbitrary natural seed value. It is conjectured that the sequence will always end in , repeating infinitely. This has been verified by computer up to very large values of , but is unproven in general. It is also not known whether this problem is decideable. This is generally called the Collatz problem The sequence

12. Collatz 3n+1 Problem Structure
    solely with the original Collatz (3n+1)/2 i conjecture, contains the main lineof argument which I believe might lead to a proof of the collatz problem.  
    http://www-personal.ksu.edu/~kconrow/
    
    Extractions: As of mid-March 2004, I've again exhausted what I have to say about the Collatz conjecture, including pages to tell about my recent discovery that there exists a whole family of analogs, (3* n j i , for small i and j =0 to infinity, to the original Collatz iteration on the integers. I've also added a page about metrics for infinity and some annotations to the links on this page to assist the reader in navigation. 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. 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. If you want a quick trip through my work, look at a 20 slide slide show which contains a few pointers to illustrative material. Always use your browser's back button to return to the slide show if you look at some auxiliary material. This first group of pages, concerned solely with the original Collatz (3

13. LookSmart - Directory - Collatz Problem
    Relations collatz problem. collatz problem Access papers, documents,and resources dedicated to the number theory collatz problem.  
    http://search.looksmart.com/p/browse/us1/us317914/us328800/us1164188/us10022111/

14. LookSmart - Directory - Unsolved Problems
    Ken Conrow Home Page Provides a collection of links to resources aboutthe collatz problem organized along the lines of a proof.  
    http://search.looksmart.com/p/browse/us1/us317914/us328800/us1164188/us10022111/

15. The Complexity Of The Collatz Problem
    The Complexity of the collatz problem The collatz problem is a verysimple, wellknown and unresolved problem of number theory.  
    http://www.geocities.com/CapeCanaveral/Lab/4430/collatz.html
    
    Extractions: The question is: does this journey always end with 1? Computers have calculated this for numbers up to millions, and they've always ended at 1. But it has never been proven it has to be so for every number. Many mathematicians have attacked the problem with no result. Legend says scientists in Los Alamos spent a good deal of their time with it, instead of working in the atomic bomb! It was even rumored it was a Russian sabotage.

16. AxyxzA.html
    It should be clear also that I am learning about sets, logic and proofwhile I am exploring the things I find about the collatz problem.  
    http://www.geocities.com/ernst_berg@sbcglobal.net/Axyxz.html
    
    Extractions: Sharing is a way of learning This is my hobby so, believe at your own risk. All data needs to be checked by you before use. Feedback is welcome. I may not know about previous works and I want to. Steve Dutch's web site has an interesting read. What Pseudoscience Tells us About Science NEWS : The page is being updated. Starting September 6, 2003 I will add what I can. Some links may not exist yet. The problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm , and Ulam's problem http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/links.htm at the Centre for Experimental and Constructive Mathematics hosts a fine example of a collection of data and information. The heart of the system [3x+y , x/2] is iteration based on a number being even or odd .

17. Collatz Problem
    exist a. collatz problem. Take any natural number m 0. n=m; repeatif (n is odd) then n=3*n+1; else n=n/2; until (n1). The conjecture  
    http://db.uwaterloo.ca/~alopez-o/math-faq/node61.html

18. Unsolved Problems
    300). collatz problem. Take any natural number m 0. n = m; repeatif (n is odd) then n = 3*n + 1; else n = n/2; until (n1).  
    http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node30.html
    
    Extractions: Next: Mathematical Games Up: Famous Problems in Mathematics Previous: Which are the 23 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 Take any natural number

19. Rechenkraft.net :: Portal
    Translate this page Projektdetails Logo. (kein Logo). Überblick. Name, collatz problem (3x+1).Kategorie, Mathematik. Ziel, Beweis der Collatz-Vermutung. Kommerziell, Nein.  
    http://217.160.138.71/project.php?id=threex

20. The Collatz Problem (3x+1)
    The collatz problem (3x+1). I was introduced to the collatz problemback in 1990 by Dr. Ashok T. Amin here in the Computer Science  
    http://hsvmovies.com/static_subpages/personal/math/collatz.html
    
    Extractions: 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:

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