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

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Mathematical Problems Prize Problems ... 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

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

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. 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

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook 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

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

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

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

The 3x+1 Problem "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/

Homepage of Keith Matthews I took early retirement from the Department of Mathematics The University of Queensland on 2nd February 2001, after an academic career of 36 years. I maintain the , which went online on 24th November 1995. email: krm@maths.uq.edu.au The Number Theory Web Mathematical Interests Mathematical Gateways to the World Wide Web Some exact arithmetic computer programs I have written: CALC : A number theory calculator program CMAT : A matrix calculator program BC number theory programs 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: , A first course in algebraic number theory (Semester 2, 2000) , A first course in number theory (Semester 2, 1999) 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 CityCat Buses 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

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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/

On the 3x + 1 problem 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. This page contains the following sections: In part 1 the problem is defined In part 2 the Glide is defined and investigated In part 3 the Delay and Residue are introduced In part 4 the Completeness and Gamma are defined In part 5 we'll discuss Class Records In part 6 Strength and Levels are introduced In part 7 Path Records are investigated In part 8 there are references to related pages The current status of the problem is given Join the distributed search for new class records! Watch the progress of the distributed search project Find pages quickly on the Site Map Latest Path Record news: In September 2003 another new Path Record was found, replacing the record found in March 2003. The record occurs at , (or ) and it reaches a maximum of which is just around 10% beyond the previous record.

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/

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

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

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List 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 "Collatz problem" is owned by akrowne view preamble View style: 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

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/

Ken Conrow Home Page Collatz 3n+1 Problem Structure 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/

@import url(/css/us/style.css); @import url(/css/us/searchResult1.css); Home IN the directory this category YOU ARE HERE Home Sciences Mathematics Number Theory Collatz Problem - Access papers, documents, and resources dedicated to the number theory Collatz problem. Directory Listings About 3x + 1 Problem and its Generalizations Jeffrey C. Lagarias writes this document on the 3x+1 problem also known as Collatz problem and the Syracuse problem. Read an introduction and generalizations. 3x+1 Problem and Related Problems Find a paper on the Collatz Problem that originally appeared in Amer. Math. Monthly in 1985. With a bibliography in PostScript format. Andrew Shapira - Image from the Collatz Problem Researcher at the Rensselaer Polytechnic Institute presents an image from the 3x+1 problem created using Fractint, a software package for generating images of fractals. Frequently Asked Questions in Mathematics - Collatz Problem Check out a document by Alex Lopez-Ortiz that includes an entry on Collatz Problem along with other number theory topics and famous problems. International Conference on the Collatz Problem International conference discussed the Collatz Problem and related topics. Download papers and abstracts of talks given during the proceedings.

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/

@import url(/css/us/style.css); @import url(/css/us/searchResult1.css); Home IN the directory this category YOU ARE HERE Home Sciences Mathematics Number Theory Unsolved Problems - Take a look at different conjectures from historical mathematicians, and join the search for answers. Directory Listings About Beal, Andrew - The Beal Conjecture and Prize Find the conjecture devised by Andrew Beal in the process of seeking Fermat's proof, and see the rules for winning the prize associated with it. Christian Goldbach Find a biography of this professor of mathematics, the story behind the Goldbach Conjecture, and a host of recommended resources. Diophantine m-tuples Browse historical and recent results on sets with the property that the product of any two elements is one less than a square. Goldbach Conjecture Research Read an analysis of several different methods of proving Goldbach's conjecture. Includes several tables and graphs illustrating examples proved computationally. Goldbach Conjecture Verification Author for this resource provides computational results, graphs, and visual examples. Check out different recommended resources. Goldbach's Conjecture - The Prime Glossary Locate a short biography on Goldbach, a description of his mathematical hypothesis, and various recommended resources.

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

The Complexity of the Collatz problem The Collatz problem is a very simple, well-known and unresolved problem of number theory. It can be expressed like this: 1. Take any integer number. 2. Divide it by 2. If the division is exact, repeat step 2. 3. If it isn't, multiply it by 3, add 1 and go to step 2. For example, if you start with 7, you'll get: 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. I haven't solved it, I've got no idea about how to solve it, but I have a good insight of how complex it can be. Let's consider a generalized version of the problem: 1. Take any Gauss integer 2. Divide it by another called a . If the division is exact, repeat step 2. 3. If it isn't, multiply it by

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

Welcome to my Hobby page The A(x)+y , x/z page The Collatz problem has interested me since 1991 It is only fitting to make a page. 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. Introduction. T here is a math problem attributed to Lothar Collatz that is the interest of many people. 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 . It is suspected that the iteration will result in the value of X looping through the same set of values after some number of iterations.

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

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

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

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.

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

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:

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