21. The 3n+1 Collatz Problem The 3n+1 collatz problem. This problem has a mysterious history andit is not easy to explain the attraction of mathematicians for it. http://www.ime.usp.br/~mane/collatz.html

The 3n+1 Collatz Problem This problem has a mysterious history and it is not easy to explain the attraction of mathematicians for it. Consider f(n)=n/2 in n is even and f(n)=(3n+1)/2 if n is odd. It is true that, for all n, the orbit of n by f ends in the cycle (1-2)? The answer for this question is unknown and a lot of problems have borned in the research of this answer (e.g. it is true that the conjecture holds for a set A of positive density? (answer: unknown). Or, if the orbit of n by f is bounded it ends in the cycle (1-2)? (answer: unknown)). We study several aspects of the dynamical system defined by function f and several generalizations of this function. Members Active in this Field (at IME-USP) Manuel Valentim de Pera Garcia - (e-mail mane@ime.usp.br Fabio Armando Tal - (email fabiotal@ime.usp.br Publications A note on the generalized 3n+1 problem - Manuel V. P. Garcia and Fabio A. Tal - Acta Arithmetica XC 3 (1999) pp. 245-250 mane@ime.usp.br Up to research fields in applied math

22. Mathenomicon.net : Reference : Collatz Problem collatz problem. noun. The collatz problem is as follows Take any x Î %Nset(); % ; If x is odd, set x to be 3x + 1. Otherwise, set x to be x/2; http://www.cenius.net/refer/display.php?ArticleID=collatzproblem

23. The Collatz Problem The collatz problem. This is one of the unsolved problems in Mathematics.You start with a positive integer. If it is even, you halve http://www.maths.ex.ac.uk/pumc/collatz.html

The Collatz problem This is one of the unsolved problems in Mathematics. You start with a positive integer. If it is even, you halve it; if it is odd you multiply by 3 and add 1. You repeat the rule on the result and keep on going, repeating the rule on that result and so on. The conjecture is that you will finish up at 1. E.g. However, nobody has managed to prove this yet. Try playing about with this simple idea and check your sequences with this form Type your starting value for the sequence

24. PlanetMath: Collatz Problem collatz problem, (Conjecture). We define the function so that. is an arbitraryseed value. collatz problem is owned by akrowne. (view preamble) http://planetmath.org/encyclopedia/142Problem.html

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports information Docs Classification News Legalese ... TODO List Collatz problem (Conjecture) We define the function so that is an arbitrary seed value. It is conjectured that the sequence will always end in , which repeats 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 sometimes called the ``hailstone sequence'' because, like a hailstone in a cloud, the values oscillate up and down. "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

25. Collatz Problem An Image from the collatz problem The intensity of a point denotes the time takento terminate. collatz problem - From Eric Weissten's World of Mathematics. http://numbersorg.com/NumberTheory/CollatzProblem/

NUMBERSorg.com Click Here for a fast ON-LINE APPLICATION- SUPER Sale Happening NOW! Search The Net Top Science Math Number Theory : Collatz Problem The Structure of the 3x + 1 Function - A paper by Peter Schorer describing a new approach to the 3x + 1 Problem and indicating a possible solution. The 3x + 1 Problem and its Generalizations - A survey article by Jeff Lagarias. The 3x+1 Problem Annotated Bibliography - By Jeffrey Lagarias, 1997. On The 3x + 1 Problem - These pages supply numerical data and propose some conjectures on this innocent looking problem. All numbers up to 29,300 * 10^12 ( ~ 26 * 2^50 ) have been checked for convergence. The Generalised 3x+1 Problem - A survey by Keith Matthews. Experiments with the 3n+1 Sequence - An online calculator by Alfred Wassermann. An Image from the Collatz Problem - The intensity of a point denotes the time taken to terminate. Collatz 3n+1 Problem Structure - Observations posted by Ken Conrow to stimulate further research. Collatz Problem - From Eric Weissten's World of Mathematics. Article with references and links. The Superset Algorithm - The "limited halting problem": finding machines that solve the halting problem for limited classes of inputs without reporting erroneous results. The Collatz problem is a special case. Software, papers and graphics.

26. Collatz Problem Information Sites Reviewed collatz problem sites, by people who know collatz problem and work withcollatz problem. collatz problem From Eric Weissten's World of Mathematics. http://numbersorg.com/NumberTheory/OpenProblems/CollatzProblem/

NUMBERSorg.com Search SPYorg.com (Not sure of spelling? Use first letters and * such as abc* or abcd* or abcde*) Match:.. All Any Format: Long Short Search Words: Top Science Math Number Theory ... Open Problems : Collatz Problem The Structure of the 3x + 1 Function - Papers by Peter Schorer describing several new approaches. The 3x + 1 Problem and its Generalizations - A survey article by Jeff Lagarias. The 3x+1 Problem Annotated Bibliography - By Jeffrey Lagarias, 1997. On The 3x + 1 Problem - These pages supply numerical data and propose some conjectures on this innocent looking problem. All numbers up to 29,300 * 10^12 ( ~ 26 * 2^50 ) have been checked for convergence. The Generalised 3x+1 Problem - A survey by Keith Matthews. Experiments with the 3n+1 Sequence - An online calculator by Alfred Wassermann. Collatz 3n+1 Problem Structure - Observations posted by Ken Conrow to stimulate further research. Collatz Problem - From Eric Weissten's World of Mathematics. Article with references and links. The Superset Algorithm - The "limited halting problem": finding machines that solve the halting problem for limited classes of inputs without reporting erroneous results. The Collatz problem is a special case. Software, papers and graphics. The 3x+1 Problem Conference Proceedings - Katholische Universit¤t Eichst¤tt, Germany; 56 August 1999. On-line proceedings and group photo.

27. The Proof Of The Collatz Problem The proof of the collatz problem. The proof by collatz function. a n /2, if an =0 (mod 2). The third step is practiced the collatz problem. Then By My Formula. http://www.angelfire.com/journal2/collatz/Proof.htm

The proof of the Collatz Problem The proof : by collatz function a n if a n (mod 2) a n n if a n (mod 2) The first step in my proof is compress the multi-function in one formula a n+1 2*s-1 +s)*a n +s , a n =s (mod 2) The second step is generated the function a n+1 +s )*a n +s , a n =s (mod 2 a n+1 +s )*a n +s +s )+s a n =s (mod 2) We can find a general formula for this expression The third step is practiced the Collatz problem Then By My Formula and we omit it because a n dosen’t equal zero And 2*s(j-i)-1 doesn’t equal zero. and The forth step we suppose that at ,a n+1 =1 if it holds a n+1 at any number less than The fifth step is analysed a n =1 and =1 it will be omit because s j- i =0 either or 1 only a n and ,s j-i =0 then a n =2 and it is right But is 1= And by Terras (1976, 1979) function t n if t n =0 (mod 2) t n+1 n if t n =1 (mod 2) We find that for all 1 step by 3t n +1 there is two steps by divide on 2 then (1/3)j= And then and Collatz conjecture be right.

28. Bookmarks 3xp1 Problem Syracuse Collatz superLagarias.html The Collatz (3x + 1) Problem Ilan Vardi http//www.mathsource.com/Content/Applications/Mathematics/0200305 collatz problem http//www http://perso.wanadoo.fr/patrick.davalan/Liens/liens_syracuse.html

3x+1, Collatz, Syracuse Problem Pages Web Livres Papiers Journaux ... Liens Avertissement : PAGES WEB The 3x+1 problem and its generalizations On the 3x+1 problem By Eric Roosendaal Glide Records Table. Delay Records Table residues page completeness page ... Super-Index of The 3x+1 Problem and its generalizations This index was automatically generated using a new tagging program written by Simon Plouffe at the CECM, Simon Fraser University. The Collatz (3x + 1) Problem Ilan Vardi Collatz Problem The Collatz (3x+1) Problem Colin Andrew Brady Campbell The 3x+1 Problem The Collatz Problem (3x+1) Evans A Criswell wondrous On The 3x + 1 Problem Unsolved problems 5x+1 and 7x+1 search results ... The Collatz (3x+1) Problem DEMOS on-line javascript Collatz trajectory calculato Mathematical mysteries - Hailstone sequences Hailstone Evaluator Embedding the 3x+1 Conjecture in a 3x+d Context Edward G. Belaga Maurice Mignotte LIVRES - BOOKS Unsolved Problems in Number Theory (Problem Books in Mathematics, Vol 1) by Richard K. Guy DOCUMENTS - PAPERS Collatz Problem Ken Monks's Faculty-Student Research Program Information A Distributed Search Program for the 3x+1 problem. Gary T. Leavens Computer Science Technical Reports A heuristic argument.

29. Collatz Problem collatz problem. A Conjecture. Let be an Integer. Then the collatz problemasks if iterating, (1). always returns to 1 for Positive . http://lib4web.lib.msu.edu/crcmath/math/math/c/c433.htm

Collatz Problem A problem posed by L. Collatz in 1937, also called the x +1 Mapping Hasse's Algorithm Kakutani's Problem ... Thwaites Conjecture , and Ulam's Problem (Lagarias 1985). Thwaites (1996) has offered a 1000 reward for resolving the Conjecture . Let be an Integer . Then the Collatz problem asks if iterating always returns to 1 for Positive . This question has been tested and found to be true for all numbers (Leavens and Vermeulen 1992), and more recently, (Vardi 1991, p. 129). The members of the Sequence produced by the Collatz are sometimes known as Hailstone Numbers Negative numbers are included, there are four known cycles (excluding the trivial cycle): (4, 2, 1), ( ), and ( ). The number of tripling steps needed to reach 1 for , 2, ... are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane's The Collatz problem was modified by Terras (1976, 1979), who asked if iterating always returns to 1 for initial integer value . If Negative numbers are included, there are 4 known cycles: (1, 2), ( ), and ( ). It is a special case of the ``generalized Collatz problem'' with , and . Terras (1976, 1979) also proved that the set of

30. AMCA: The Relation Between Collatz Conjecture And Primes Presented By Murad A. A the wellknown conjecture ,Collatz conjecture One of many conjectures still withoutproof a problem posed by L. Collatz in 1937 The collatz problem asks if http://at.yorku.ca/cgi-bin/amca/caft-16

AMCA Document # caft-16 Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics December 21-24, 2000 University of Craiova Craiova, Romania Organizers Minh Perez (American Research Press, Rehoboth, Box 141, NM 87301, USA), Vasile Seleacu (University of Craiova, Department of Mathematics, Craiova, Romania) View Abstracts Conference Homepage The relation between collatz conjecture and primes by Murad A. AlDamen University of Jerash Date received: October 4, 2000 Atlas Mathematical Conference Abstracts

31. Science/Math/Number_Theory/Collatz_Problem The Collatz, Syracuse or 3x+1 problem / Science /Math / Number_Theory / collatz problem. The collatz problem is a special case.......Search Category http://www.science-and-research.com/Science/Math/Number_Theory/Collatz_Problem/

Search: Category Description: The Collatz, Syracuse or 3x+1 problem Science Math Collatz Problem The Structure of the 3x + 1 Function A paper by Peter Schorer describing a new approach to the 3x + 1 Problem and indicating a possible solution. URL: http://www.occampress.com The 3x + 1 Problem and its Generalizations A survey article by Jeff Lagarias. URL: http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html The 3x+1 Problem Annotated Bibliography By Jeffrey Lagarias 1997. URL: http://www.research.att.com/~jcl/3x+1.html On The 3x + 1 Problem These pages supply numerical data and propose some conjectures on this innocent looking problem. All numbers up to 29 300 * 10^12 ( ~ 26 * 2^50 ) have been checked for convergence. URL: http://personal.computrain.nl/eric/wondrous/ The Generalised 3x+1 Problem A survey by Keith Matthews. URL: http://www.maths.uq.edu.au/~krm/survey.pdf Experiments with the 3n+1 Sequence An online calculator by Alfred Wassermann. URL: http://did.mat.uni-bayreuth.de/Fun/3np1_e.html An Image from the Collatz Problem The intensity of a point denotes the time taken to terminate.

32. Www-irma.u-strasbg.fr/irma/seminaires/autres/ACP-2001.shtml Similar pages IRMA Strasbourg Publication 2000 Belaga,Maurice Mignotte, (fichier Postscript compressé, 792 Ko) Cyclic Structureof Dynamical Systems Associated with $3x+d$ Extensions of collatz problem.. http://www-irma.u-strasbg.fr/irma/seminaires/autres/ACP-2001.shtml

33. Introduction To Artificial Intelligence Selmer Bringsjord Hyperproof file.); A picture of the collatz problem; More on the CollatzProblem and Related Problems; Is English Compositional? (picture http://www.rpi.edu/~brings/intai.html

The Intro to AI Show Your Host, Selmer Bringsjord and Selmer as.... Spock with your two Co-Hosts The Terminator 7+MB quicktime movie Selmer himself Meet the Cast Video Highlights Handouts ... Supporting Links Handouts Syllabus ( html dvi postscript pdf Practice Exams For Practice Final Summer 99 postscript pdf Practice Midterm Summer 99 ( html postscript dvi pdf Practice Final Su 97 ( html postscript dvi pdf Practice Midterm Su 97 ( html postscript pdf Midterm Su 97 ( html dvi postscript pdf Problems/Puzzles for Smart Agents Godelian Puzzle 1 ( html dvi postscript Godelian Puzzle 2 ( html dvi postscript The Mystery of the Missing Peraflop Machine ( html postscript The Dreadsbury Mansion Mystery ( documentation graphic problem (hqx) The Wise Man Puzzle ... possible formalizations The Lottery Paradox ( html postscript The Schubert Steamroller Problem ( html dvi ps Projects Project 3 Finish Shapiro's ELIZA, i.e., (p1). Due by 5pm on day of final exam for studio audience. Due on same day as final for home audience (emailed to Selim as ASCII attachment).

34. Www.math.niu.edu/~rusin/known-math/99/collatz_undec seem to recall that Conway may have established this fact.) Conway defined a familyof problems, a natural generalization of the collatz problem, and showed http://www.math.niu.edu/~rusin/known-math/99/collatz_undec

From: gerry@mpce.mq.edu.au (Gerry Myerson) Subject: Re: Is Collatz Conjecture (3N+1 Problem) Undecidable? Date: Wed, 30 Jun 1999 13:05:01 +1100 Newsgroups: sci.math Keywords: Conway's generalizations include some undecideable problems In article

35. Www.math.niu.edu/~rusin/known-math/98/collatz_hide From Bill Daly billdaly@interport.net Newsgroups sci.math.research,sci.mathSubject Reformulation of collatz problem Date Wed, 4 Mar 1998 121004 http://www.math.niu.edu/~rusin/known-math/98/collatz_hide

From: "Bill Daly" , Bill Daly

36. Sci.math FAQ: Unsolved Problems collatz problem * Goldbach's conjecture * Twin primes conjecture _Names of large numbers http://isc.faqs.org/faqs/sci-math-faq/unsolvedproblems/

sci.math FAQ: Unsolved Problems Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76LD.Fnt@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math hv@cix.compulink.co.uk (Hugo van der Sanden): To the best of my knowledge, the House of Commons decided to adopt the US definition of billion quite a while ago - around 1970? - since which it has been official government policy. dik@cwi.nl (Dik T. Winter): The interesting thing about all this is that originally the French used billion to indicate 10^9, while much of the remainder of Europe used billion to indicate 10^12. I think the Americans have their usage from the French. And the French switched to common European usage in 1948. gonzo@ing.puc.cl alopez-o@barrow.uwaterloo.ca By Archive-name By Author ... Help Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update March 05 2003 @ 01:20 AM

37. F. Conjectures (Math 413, Number Theory) A collection of easily stated conjectures which are still open. Each conjecture is stated along with Category Science Math Number Theory Open Problems...... Conj All integers greater than 1 have occur as multiplicities. (Sierpinski) RefsUnsolved Problems in Number Theory, Guy, 1994, Sect B.37. The collatz problem. http://www.math.umbc.edu/~campbell/Math413Fall98/Conjectures.html

F. Conjectures Number Theory, Math 413, Fall 1998 A collection of easily stated number theory conjectures which are still open. Each conjecture is stated along with a collection of accessible references. The Riemann Hypothesis Fermat Numbers Goldbach's Conjecture Catalan's Conjecture ... The Collatz Problem The Riemann Hypothesis Def: Riemann's Zeta function, Z(s), is defined as the analytic extension of sum n infty n s Thm: Z( s )=prod i infty p i s , where p i is the i th prime. Conj: The only zeros of Z( s ) are at s s Thm: The Riemann Conjecture is equivalent to the conjecture that for some constant c x )-li( x c sqrt( x )ln( x where pi( x ) is the prime counting function. Refs: Unsolved Problems Elementary Number Theory http://www.utm.edu/research/primes/notes/rh.html (notes in Chris Caldwell's Prime Pages) http://match.stanford.edu/rh/ (Dan Bump's notes) Def: n is perfect if it is equal to the sum of its divisors (except itself). Examples are 6=1+2+3, 28, 496, 8128, ... Def: The n th Mersenne Number, M n , is defined by M n n Thm: M n is prime implies that n n is perfect. (Euclid)

38. Numcom08 NUMBERS AND COMPUTERS (8) by Albert N. Debono THE collatz problem. Takeany positive number. If it is odd, multiply it by three and add one. http://www.eng.um.edu.mt/~andebo/numbers/numcom08.htm

NUMBERS AND COMPUTERS (8) by Albert N. Debono THE COLLATZ PROBLEM Take any positive number. If it is odd, multiply it by three and add one. If it is even divide it by two. Repeat the process on the new number so obtained and keep on repeating this procedure. Starting with the number 3 we get: 3, 10, 5, 16, 8, 4, 2, 1. If we keep on repeating the procedure after the 1 we would of course get 4, 2, and 1 again ad infinitum i.e. we entered a loop. Thus the number 3 is brought down to 1 in 7 steps and reaches a maximum of 16. If we start with the number 27 instead of 3 we find that we need 111 steps to come down to 1 and a maximum of 9232 will be reached in the sequence. Write a program on your micro which when given a number will output the number of steps required to bring it down to 1 and also the maximum reached. The output could look like this: 27 closes in 111 steps and reaches a maximum of 9232 255 closes in 47 steps and reaches a maximum of 13120 447 closes in 97 steps and reaches a maximum of 39364 It is interesting to see a graphical representation of these sequences by writing a program on your micro so that the numbers generated in the sequence are plotted (vertically) against the step number (horizontally) and the resulting points (joined by straight lines) shown on the monitor.

39. Artslist Numbers and Amicable Pairs 4. The Prime Numbers 5. Aliquot Sequences 6. Abundantand Deficient Numbers 7. Kaprekar Numbers 8. The collatz problem 9. Mersenne http://www.eng.um.edu.mt/~andebo/numbers/artslist.htm

LIST OF ARTICLES 1. The Persistence of a Number 2. Palindromes and the 196 Problem 3. Perfect Numbers and Amicable Pairs 4. The Prime Numbers ... Back to my Home Page

40. Math Forum - Ask Dr. Math What's the difference between problems like Squaring the Circle andGoldbach's Conjecture or the collatz problem? Drexel dragon http://mathforum.org/library/drmath/view/52540.html

Associated Topics Dr. Math Home Search Dr. Math Unsolvable and Unsolved Problems Date: 02/19/98 at 18:16:57 From: D-Jitsu Subject: 11th Grade Research Paper Dear Dr. Math, I need help getting started on a research paper, as well as on doing the project. The topic of my report is "The Impossible and Unsolved in Mathematics (Goldbach's Conjecture, Squaring the Circle, etc.). I was just wondering if I could get some references, tips, or ideas on how to get started. This is for a pre-calculus class. http://ug.cs.dal.ca/~campbell/colllink.html http://mathforum.org/dr.math/ Associated Topics High School History/Biography High School Number Theory Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

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