61. Elementary Number Theory - Kenneth H. Rosen College in Dublin, Ireland can be found at http//www.spd.dcu.ie/johnbcos/Maple_3rd_year.htmYou can find a Maple worksheet on the collatz problem at http//www http://www.aw.com/rosen/resources_app.html

Annotated Web Links APPENDICES Return to Annotated Web Links Home Appendix A Axioms for the Set of Integers Page 515 You can learn more about the Peano axioms at http://www.torget.se/users/m/mauritz/math/num/peano.htm (Peano's Axioms) Appendix B Binomial Coefficients Page 521 Biographical information about Blaise Pascal can be found at the MacTutor History of Mathematics Archive at http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pascal.html (Pascal) Appendix C Using Maple and Mathematica for Number Theory Page 527 The Maple home page is a good place to starting learning more about Maple: http://www.maplesoft.com (Waterloo Maple: Home) Page 528 Maple worksheets written by John Cosgrave for a course in number theory and cryptography at St. Patrick's College in Dublin, Ireland can be found at http://www.spd.dcu.ie/johnbcos/Maple_3rd_year.htm

62. Internet Center For Mathematics Problems collatz problem. Famous Problems in Mathematics The Four Colour Theorem;The Trisection of an Angle. Which are the 23 Hilbert Problems? http://www.mathpropress.com/mathCenter.html

Internet Center for Mathematics Problems On this page, we try to identify and list all sources of mathematics problems on the Internet and related information. Please advise us if you know of any sources that we are missing. Table of Contents Problem Columns from Mathematics Journals The Fibonacci Quarterly Mathematics and Informatics Quarterly (coming soon) Missouri Journal of Mathematical Sciences Problem Columns on the World Wide Web Mathematics Problems on the World Wide Web Newsgroups ... Sign-in We invite you to visit the MathPro Press home page Problem Columns from Mathematics Journals Many mathematics journals contain problem columns. They invite readers to send in solutions to the problems. The best solution submitted gets printed in a future issue along with the names of all the solvers. We list below only a few of the journals that are currently running problem columns. You are welcome to try your hand at solving these problems. Solutions should be sent to the appropriate problem column editor ( not to MathPro Press).

63. 3x+1.html By Jeffrey Lagarias, 1997.Category Science Math Number Theory Open Problems collatz problem......Jeffrey C. Lagarias 3x+1 problem and related problems. The 3x+1 problem and itsgeneralizations , Jeffrey C. Lagarias, Amer. Math. Monthly 92 (1985) 323. http://www.research.att.com/~jcl/3x 1.html

64. More Hailstones... name. You may also see it referred to as the Syracuse problem, thecollatz problem or simply as the 3n+1 conjecture. A conjecture http://plus.maths.org/issue2/news/hail/

PRIME NRICH PLUS Current Issue ... Subject index Issue 22: Jan 03 Issue 21: Sep 02 Issue 20: May 02 Issue 19: Mar 02 Issue 18: Jan 02 Issue 17: Nov 01 Issue 16: Sep 01 Issue 15: Jun 01 Issue 14: Mar 01 Issue 13: Jan 01 Issue 12: Sep 00 Issue 11: Jun 00 Issue 10: Jan 00 Issue 9: Sep 99 Issue 8: May 99 Issue 7: Jan 99 Issue 6: Sep 98 Issue 5: May 98 Issue 4: Jan 98 Issue 3: Sep 97 Issue 2: May 97 Issue 1: Jan 97 More hailstones... In the last issue of PASS Maths we presented the hailstone sequence: an unsolved mathematical mystery. Many of our readers have tried out our hailstone sequence generator and others have asked for more information about the problem or simply why it is of interest. The hailstone sequence is not the problem's only name. You may also see it referred to as the Syracuse problem the Collatz problem or simply as the 3n+1 conjecture . A conjecture is simply a mathematical statement which has not yet been proved. To our knowledge the principal conjecture "the sequence terminates at the value 1 for all starting values greater than 0" is still unproved, despite rumours to the contrary! To try out even longer sequences (using larger starting values than those offered by our own hailstone generator) you could try combining your Mathematical skills with your German and read Alfred Wassermann's page "Experiment mit der 3n+1 Folge" (literally "Experiment with the 3n+1 sequence").

65. Collatz Problem -- From MathWorld Similar pages collatz.html Abstract Worksheet performs number of iterations in collatz's problem,ie, the 3x+1 conjecture. Application Areas/Subjects Number Theory. http://www.astro.virginia.edu/~eww6n/math/CollatzProblem.html

Number Theory Sequences 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 the 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 ). Of these, the numbers of tripling steps are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane's ), and the number of halving steps are 0, 1, 5, 2, 4, 6, 11, 3, 13, ... (Sloane's ). The smallest starting values of that yields a Collatz sequence containing n = 1, 2, ... are 1, 2, 3, 3, 3, 6, 7, 3, 9, 3, 7, 12, 7, 9, 15, 3, 7, 18, 19, ... (Sloane's

66. Maple Application Center Entry Type, Code, worksheet, TeX document; share. Summary, Performs number of iterationsin collatz's problem, ie, the 3x+1 conjecture. Author(s), Gaston Gonnet. http://www.mapleapps.com/maplelinks/share/collatz.html

67. Mathematica Information Center: The Collatz (3x + 1) Problem Title, Downloads, The collatz (3x + 1) problem, Author, Keywords, Examples, collatzproblem, 3x+1 problem, collatz, TotalStoppingTime, numerical sequences, Downloads, http://library.wolfram.com/database/Demos/153/

All Collections Articles Books Conference Proceedings Courseware Demos MathSource: Packages and Programs Technical Notes Title The Collatz (3x + 1) Problem Author Ilan Vardi Organization: Wolfram Research, Inc. Description This package computes the iterates of the Collatz map: x -> x/2, if x is even; x -> (3x+1)/2, if x is odd, until an iterate reaches one of the four known cycles (the program runs on positive and negative integers). Subject Mathematics Number Theory Keywords Examples, Collatz problem, 3x+1 problem, Collatz, TotalStoppingTime, numerical sequences Downloads Collatz.m (4 KB) - Mathematica Package Files specific to Mathematica 2.2 version: Collatz.m (4 KB) - Mathematica Package

68. Problem 1 — The Collatz Sequence problem 1 — The collatz Sequence. An algorithm given by Lothar collatzproduces sequences of integers, and is described as follows http://www.acm.inf.ethz.ch/ProblemSetArchive/B_US_NorthCen/1998/prob1.htm

Problem 1 — The Collatz Sequence An algorithm given by Lothar Collatz produces sequences of integers, and is described as follows: Step 1: Choose an arbitrary positive integer A as the first item in the sequence. Step 2: If A = 1 then stop. Step 3: If A is even, then replace A by A / 2 and go to step 2. Step 4: If A is odd, then replace A by 3 A + 1 and go to step 2. It has been shown that this algorithm will always stop (in step 2) for initial values of A as large as 10 , but some values of A encountered in the sequence may exceed the size of an integer on many computers. In this problem we want to determine the length of the sequence that includes all values produced until either the algorithm stops (in step 2), or a value larger than some specified limit would be produced (in step 4). The input for this problem consists of multiple test cases. For each case, the input contains a single line with two positive integers, the first giving the initial value of A (for step 1) and the second giving L, the limiting value for terms in the sequence. Neither of these, A or L, is larger than 2,147,483,647 (the largest value that can be stored in a 32-bit signed integer). The initial value of A is always less than L. A line that contains two negative integers follows the last case. For each input case display the case number (sequentially numbered starting with 1), a colon, the initial value for A, the limiting value L, and the number of terms computed.

69. About "Collatz 3n+1 Problem Structure" collatz 3n+1 problem Structure. Library Home Full Table of Contents Suggesta Link Library Help Visit this site http//wwwpersonal.ksu.edu/~kconrow/. http://mathforum.org/library/view/17463.html

Collatz 3n+1 Problem Structure Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www-personal.ksu.edu/~kconrow/ Author: Ken Conrow Description: Pages of observations posted to stimulate further research. Levels: College Research Languages: English Math Topics: Number Theory Suggestion Box Home The Math Library ... Search http://mathforum.org/ webmaster@mathforum.org

70. From The North Central Regionals 1998 - Problem 1 — The Collatz Sequence From the North Central Regionals 1998 problem 1 — The collatz Sequence. Javasolution. Solve the problem in one file calling it collatz.java . http://www.ug.cs.usyd.edu.au/~acmipc/week09questfirst.html

From the North Central Regionals 1998 - Problem 1 — The Collatz Sequence (Please follow the notes given at the end of this document for submitting your solution.) An algorithm given by Lothar Collatz produces sequences of integers, and is described as follows: Step 1: Choose an arbitrary positive integer A as the first item in the sequence. Step 2: If A = 1 then stop. Step 3: If A is even, then replace A by A / 2 and go to step 2. Step 4: If A is odd, then replace A by 3 A + 1 and go to step 2. It has been shown that this algorithm will always stop (in step 2) for initial values of A as large as 10 , but some values of A encountered in the sequence may exceed the size of an integer on many computers. In this problem we want to determine the length of the sequence that includes all values produced until either the algorithm stops (in step 2), or a value larger than some specified limit would be produced (in step 4). The input for this problem consists of multiple test cases. For each case, the input contains a single line with two positive integers, the first giving the initial value of A (for step 1) and the second giving L, the limiting value for terms in the sequence. Neither of these, A or L, is larger than 2,147,483,647 (the largest value that can be stored in a 32-bit signed integer). The initial value of A is always less than L. A line that contains two negative integers follows the last case. For each input case display the case number (sequentially numbered starting with 1), a colon, the initial value for A, the limiting value L, and the number of terms computed.

71. Billstein - Problem Solving Approach To Math Chapter 1 -- TI Programs These TI82 and TI- 73 programs automate collatz's problem. Thatis, given a positive integer N, the number is either halved if http://occawlonline.pearsoned.com/bookbind/pubbooks/billstein_awl/chapter1/custo

TI Calculator Downloadable Programs The programs below are for the TI-73, TI-81, TI-82, and TI-83 graphing calculators from Texas Instruments and were created by Aron Cummings of Washington State University and Michael Lloyd of Henderson State University. Using GraphLink software from Texas Instruments, you can download a program from your computer to your graphing calculator. If this is the first time you are downloading a TI calculator program, or if you do not have GraphLink software, you can find help on the instructions page located on the Texas Instruments Web site. In the list below, click on the name of the program to download it. Choose the .73, .82, .83 or .txt file extension, depending on the graphing calculator that you are using. Files Description BASECONV.83p BASECONV.txt This TI-83 program converts positive numbers between bases. (The base cannot be more than 36.) You cannot enter expressions like "1/3". Input A = starting base B = ending base COINTOSS.82p COINTOSS.73p These TI-82 and TI-73 programs simulate the tossing of N coins T times. The program gives the number of times in the T tosses that no head, 1 head, 2 heads, . . . , and all N coins showed a head. Inputs N (the number of coins to be tossed at a time) and T (the number of times the N coins are tossed).

72. Math 696 The 3x+1 Problem There is a famous unsolved problemoften called the 3x+1 problemaboutthe iterates of the collatz function. Is it the case that http://www.math.tamu.edu/~harold.boas/courses/math696/Maple-3x 1.html

73. Adept Scientific Plc - The Technical Computing People Subject Number of iterations in collatz's problem. Date AbstractThe 3n+1 sequence has probably consumed more CPU time than http://www.adeptscience.co.uk/products/mathsim/maple/apps/subcategory/17/Number

Adept Store Join My Adept International Sites Welcome Products Buy Online Downloads ... My Adept My Status: You are not logged in Advanced Search About Us Contact Us Press Room ... System Requirements Downloads Brochures Tech Support Training ... Number of iterations in Collatz's problem Subject: Number of iterations in Collatz's problem Date: Abstract: The 3n+1 sequence has probably consumed more CPU time than any other number theoretic conjecture. The reason being that its statement is so simple, that most amateurs will feel compelled to write programs to test it. This sequence, attributed to Lothar Co Animations: Version: Maple 5 View Document: Download Worksheet: Download Code: While every effort has been made to validate the solutions in these worksheets, Adept Scientific plc are not responsible for any errors contained and are not liable for any damages resulting from the use of this material. Contact the Maple Team Buy Maple Now View Maple Pricing Download a Brochure ... Request a Demo Learn more about Maple Maple 8 Overview Maple 8 functionality MapleNet Reinvent how you explore, teach and share mathematics. Explore...

74. Hardware Games Software Wireless Sci-Tech Entertainment Legal One of the greatest numbers subjected to the collatz conjecture that had its andStopping Time RecordHolders for the 3x+1 problem Computational Results . http://www.infosatellite.com/news/2003/01/p070103collatz.html

sections Hardware Games Software Wireless Sci-Tech Entertainment Legal Security newsletter x e-mail this to a friend others Archive Contact Us Top Stories Archive The latest news on your website for free! Advertise InfoSatellite Jobs Terms of Use/Privacy Corrections InfoSatellite.com / News The Collatz Conjecture By Pedro Gomes InfoSatellite.com January 07, 2003 Let's write it in the style of the old Basic programming language: 10 Pick any positive integer n. 20 If n is even, divide it by two; if it is odd, multiply it by three and add one. 30 If n = 1, stop; else go back to step 2. For the first digits, results are: That's the Collatz conjecture, also known as the 3n+1 conjecture, the Ulam conjecture or the Hailstone sequence, to which names Jeffrey C. Lagarias, which also discusses more deeply its implications at cecm.sfu.ca

75. NetBSD Problem Report #741: Ftp Client Mis-does SYST Query (I haven't done a fix myself this time.) der Mouse mouse@collatz.mcrcim.mcgill.edu tofigure out whether another command or two are broken by the same problem. http://www.netbsd.org/cgi-bin/query-pr-single.pl?number=741

76. The Drills To Play The problem was proposed Gardner, p. 203 by Lothar collatz and is known as thecollatz Conjecture, but also as the 3X + 1 problem and is often associated http://www.cut-the-knot.com/ctk/March2001.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Cut The Knot! An interactive column using Java applets by Alex Bogomolny The Drills To Play March 2001 John Dawson second Hilbert problem Third, it's OK to be mistaken. It's good to keep one's mind open while pursuing an idea. Who knows? The idea may metamorphose into its opposite as the result of one's efforts. But let's return to repetition and practice. Are there repetitive activities that go beyond (mindless would be a usual adjective) memorization? Why, there's a great deal of them of course. Repetition is a life line of computational mathematics. Iterative processes serve one example. Given a number and a set of rules to be performed that result in another number. Apply the rules to the new number and get a third one, and so on. Iterations may apply to a group of numbers or other mathematical objects. See, for example, the Candy game and Integer Iterations on a Circle . Following are three additional examples. Start with a number and count the number E of even and the number O of odd digits [ Ecker ]. Write them down next to each other following by their sum

77. KOTANI Lab Human Chess DBGK; collatz's problem or 3x+1 problem calculated to 100,000,000,000,000by Fujinami, and then 105,500,000,000,000 by Kawano, and finally 2000 http://www.tuat.ac.jp/~kotani/

General Index !!£³X¡Ü£±ÌäÂ긽ºß¤Îµ­Ï¿ 2000û!! £³X¡Ü£±ÌäÂê Kotani Laboratory General Home Page in English Yoshiyuki Kotani ... PUZZLE WORLD

78. P Andaloro's Abstract collatz's problem The Venus Fly Trap of Mathematics. By Dr. Paul Andaloro AssistantProfessor Department of Mathematics Kent State University (Stark Campus) http://www.ma.iup.edu/calendar/AY2001-02/Andaloro_Abstract.html

Collatz's Problem - The Venus Fly Trap of Mathematics By Dr. Paul Andaloro Assistant Professor Department of Mathematics Kent State University (Stark Campus) Dr. Andaloro will be presenting a talk on Number Theory. More specifially the 3x +1 problem. The presentation will be a combination of an expository talk and presenting some current research. Back to the Colloquia Calendar

79. Optimal Bounds For The Length Of Rational Collatz Cycles - Halbeisen, Hungerbuhl Correct) Users who viewed this document also viewed More All 0.1 The 3n+1 CollatzProblem and Functional Equations Berg, Meinardus (1995) (Correct) 0.1 http://citeseer.nj.nec.com/halbeisen97optimal.html

Optimal Bounds For The Length Of Rational Collatz Cycles (1997) (Make Corrections) (1 citation) Lorenz Halbeisen, Norbert Hungerbühler Home/Search Context Related View or download: berkeley.edu/~halbeis/psf/collatz.ps Cached: PS.gz PS PDF DjVu ... Help From: berkeley.edu/~halb publications (more) Homepages: L.Halbeisen HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: . We consider the arithmetics of Collatz cycles in Q[(2)]. In particular, we prove optimal estimates for the length of a cycle in terms of its minimum. As an application, we derive an improved version of Eliahou's criterion, and we show that the length of (integer) Collatz cycles which do not contain 1, is at least 102 225 496 provided the Collatz conjecture is verified for all initial values x 212 366 032 807 211. 1. Introduction For x 2 R let g (x) = x 2 and g 1 (x) = 3x+1 2 . Let... (Update) Context of citations to this paper: More ...no integer is known to diverge to 1, and no cycle other than the ones above has been found. By a recent theorem of Halbeisen and Hungerbuhler [1997] , any extra cycle of positive integers must contain more than 10 8 numbers (even when counting any 3n 1 step and the Cited by: More The n+1-Problem and Holomorphic Dynamics - Letherman, Schleicher, Wood

80. Informatik A Translate this page ISBN 0-201-61586-X. 267 + xii pp. US $24.95. K H. Klaeren, Vom problem zum Programm(2. Aufl.), BG Teubner K1 DE Knuth, The Art of Computer Programming http://www.mathematik.uni-bielefeld.de/~rehmann/Inf-A/

Informatik A Informatik-Einführung für Wirtschaftsmathematiker, Sommersemester 2002 Prof. Dr. Ulf Rehmann Sprechstunde: Mi 11-12, V5-239 oder n.V. (Tel. 106-5039) 24 52 01 Vorlesung: Di 14 - 16, Fr 8 - 10 in H 3 Informationen zu C und C++ Die Vorlesung umfaßt an sich drei Semesterwochenstunden, sie wird allerdings 4-stündig gelesen mit Ausnahme der Termine 19. April, 10., 14., 17.. 31. Mai, 25. und 28 Juni. Die Vorlesung gibt für Wirtschaftsmathematiker und andere eine Einführung in die Grundlagen der Informatik. Gegenstände sind die Grundbegriffe der Programmierung, Algorithmen und Datenstrukturen. Als Programmiersprache werden die Sprachen C und C++ benutzt. Die notwendigen Kenntnisse über C, C++ (und das Betriebssystem Unix) können in dem einführenden Programmierkurs erworben werden. http://www.cplusplus.com/ .mailcap in Ihrem Home-Verzeichnis (unter dem Namen .mailcap) ablegen. 01: Euklidischer Algorithmus, Newtonsche Näherung, Collatz's Problem, Turm von Hanoi. Weitere Informationen: Al'Khwarizmi Euklid Newton Zu Collatz's Problem ... 03: b-adische Darstellung, Bit-Codierung von Fliesskommazahlen, Hornersches Schema, elementare Datentypen Weitere Informationen: Fibonacci 04: Zeitbedarf von Rechenoperationen, Fibonacci-Zahlen: Exponentielle, lineare, logarithmische Implementation.

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