41. The Math Forum - Math Library - Number Theory The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Includes An image from the collatz problem, including the Fractint code; http://mathforum.org/library/topics/number_theory/

Browse and Search the Library Home Math Topics : Number Theory Library Home Search Full Table of Contents Suggest a Link ... Library Help Subcategories (see also All Sites in this category Algebraic Number Theory Analytic Number Theory Sequences and Sets Fibonacci Sequence Selected Sites (see also All Sites in this category Continued Fractions: an Introduction - Adam Van Tuyl A brief introduction to the field of continued fractions, including some basic theory about the subject; the history of continued fractions, tracing some of the major developments in the field in the past 2500 years; some interactive applications that demonstrate the uses of continued fractions and let you calculate them; and the resources used in creating this site, including a bibliography and links to other sites on the Web. more>> Fermat's Last Theorem - MacTutor Math History Archives Essay describing Fermat's theorem with links to mathematicians such as Sophie Germain, Legendre, Dirichlet, Shimura and Taniyama, etc., from its inception through Andrew Wiles' proof, with another web site and 17 references (books/articles). more>> Number Theory - Dave Rusin; The Mathematical Atlas

42. About "The 3x+1 Problem And Its Generalizations (Organic Mathematics Proceedings , The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Kakutani s problem, Hasse s algorithm, and Ulam s problem,...... http://mathforum.org/library/view/3091.html

The 3x+1 problem and its generalizations (Organic Mathematics Proceedings) Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.cecm.sfu.ca/organics/papers/lagarias/ Author: Jeff Lagarias Description: The 3x+1 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 3n+1 and even integers n to n/2. The 3x+1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1. This extensive paper previously appeared in the American Mathematical Monthly. Levels: College Research Languages: English Resource Types: Articles Math Topics: Number Theory Home The Math Library Quick Reference ... Help http://mathforum.org/

43. Collatz Problem Was Solved And Published: June 29, 2003 : Application HENs Semin http://www.crt.or.jp/~kokochi/CollatzProblem.htm

this page cannot be displayed. Another page for printing So, click here , please.

44. F. Conjectures (Math 413, Number Theory) This conjecture is variously referred to as the collatz problem (for the original worker in the field), the Syracuse problem, or the 3 x +1 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. Thm: The only zeros of Z( s ) are at s s Conj: The only zeros of Z( s ) are at s =-2, -4, -6, ... and on the line Re( 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

45. CABC - Collatz (3x+1) Problem The collatz problem is referred to in the literature as the 3x+1 problem, the Syracuse The collatz problem concerns the behaviour of a simple function, http://www.drewcampbell.ca/collatz/

history references links last modified: May, 2002 - comments: cabcampbell@hotmail.com The Collatz (3 x +1) Problem The Collatz problem is referred to in the literature as the 3x+1 problem, the Syracuse problem, Hasse's algorithm, Ulam's problem and Kakutani's problem. It was believed to have been first posed by Lothar Collatz in 1930 while he was still a student, and it's many names are an indicator of the many unsucessful attempts to solve it. The Collatz problem concerns the behaviour of a simple function, referred to as the Collatz function. The function T on the postive integers (excluding zero) defined by T( x x /2 if x is even, and T( x x +1)/2 if x is odd is one of several equivalent ways to write the Collatz function. We can iterate functions by continually applying the function to its output. Iterating T for some initial x , we create a sequence of integers for x which we call the Collatz trajectory of x Mathematically, we can define the i th iterate of T( x ) (the i th term in the Collatz trajectory) as T^ i x ) where ^ is an ASCII representation for superscript and where T^( i x ) = T(T^ i x )), T^0(

46. Practical Foundations Of Mathematics The collatz problem asks whether p is surjective. So both in the collatz problem and in logic, the image of one recursive structure in another may be an http://www.cs.man.ac.uk/~pt/Practical_Foundations/html/s62.html

Practical Foundations of Mathematics Paul Taylor Well Formed Formulae Free algebraic theories provide a useful ``scaffolding'' which can be used during the building of more complicated linguistic structures, such as dependent type theory in Chapters VIII and IX . In this section we shall describe the recursive aspects of arguments about such structures, which are for example used in the construction of the interpretation functor [[-]]: Cn L S In practice, additional side-conditions are required of the terms which are to be admitted to the language. Some of these, such as the number of arguments taken by each operation-symbol, can be enforced in advance, but others must be stated by simultaneous recursion together with the expressions themselves. The terms which do satisfy the conditions are traditionally known as wffs well formed formulae D EFINITION 6.2.1 A wff-system is a set X of terms for a free theory ( W ar ) such that if r u X j X for all j ar r ]. Therefore parse X TX is a total function on X , and is injective (since ev is a partial inverse).

47. ScienceDaily -- Browse Topics: Science/Math/Number_Theory/Open_Problems/Collatz_ Collatz 3n+1 Problem Structure Observations posted by Ken Conrow to stimulate further collatz problem - From Eric Weissten s World of Mathematics. http://www.sciencedaily.com/directory/Science/Math/Number_Theory/Open_Problems/C

@import "/styles/navbar.css"; @import "/styles/tabStyles.css"; Set home page Bookmark site Add search Latest News ... Email to friend Text Size A A A Front Page ... Open Problems : Collatz Problem Subtopics Search Google: Collatz Problem News Books Links ... Internet-based Stroke Exam Speeds Treatment In Rural Areas (September 16, 2005) full story Memory Loss In Older Adults Due To Distractions, Not Inability To Focus (September 12, 2005) full story High-performance Computing May Improve Combustion Efficiency (September 11, 2005) full story Transplanting Animal Organs Could Soon Be A Reality (September 10, 2005) full story Nanocoating Could Eliminate Foggy Windows And Lenses (August 30, 2005) full story Physicists Describe A New Mechanism For Metallic Magnetism (August 26, 2005) full story Rate Of Cellular Energy Production Lower In Persons At Risk For Type 2 Diabetes (August 26, 2005) full story Type 2 Diabetes: Problems In The Furnace (August 24, 2005) full story [ More news about Collatz Problem Numerical Treatment of Eigenvalue Problems (International Series of Numerical Mathematics) Optimization Problems (Applied Mathematical Sciences Ser. ; Vol. 17)) by: Lothar Collatz Numerical Treatment of Eigenvalue Problems: Workshop in Oberwafach, February 25-March 3, 1990/Numerische Behandlung Von Eigenwertaufgaben : Tagung (International ... of Numerical Mathematics = Internationa)

48. CS1 -- Fall 2004 -- Midterm Exam A major unsolved problem in mathematics is the collatz problem (named after We will not try to solve the collatz problem here (though anyone who does http://www.cs.caltech.edu/courses/cs1/midterm/midterm_exam.html

CS 1 Fall 2004 Midterm Exam Assigned: Thursday, October 28, 2004 Due: Tuesday, November 2, 2004, 17:00:00 Preliminaries This exam is due Tuesday, November 2, at 5pm . Please submit your text answers electronically to cs1man assignment midterm The time limit on this exam is four hours . We expect that it should take, on average, about three hours to complete. This is a closed-book exam. This is a no-reference exam, with one exception: you may consult a copy of your lab submissions, as well as any responses from your TA to those submissions, during this exam. This is a closed-interpreter exam. You may not use DrScheme or any other interpreter. Feel free to type up your exams in any plain text editor of your choice. Do not submit code in non-plaintext form; Word documents and PDFs are good examples of non-plaintext formats. Do try to make your code fit into some reasonable number of character-columns, like 80. A great way to test all this is to execute " more mymidtermfile " at the CS cluster commandline. If you can read your results then we can read your results, too! Do seek out an editor that matches parentheses if you prefer, but know that we are not grading you on whether your code is "perfect;" we have Scheme interpreters to tell us that. Indeed, we are looking for correctness at a higher level. A missing parenthesis here or there will not affect your score.

49. Mathematical Problems - Problem Solving - Mathematical Competitions 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, http://mathres.kevius.com/problem.html

Mathematical Problems - Problem Solving Mathematical Competitions not a complete list, only what I happened to see... 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 by Jeff Lagarias 21st Century Problem Solving Solutions to solving word problems across the curriculum. (SureMath - hawaii.edu) MATHCOUNTS Foundation Algebra Story and Word Problems A collection of simple algebra word problems (SureMath: 21st Century Problem Solving) Algebrator - an Algebra Problem Solver for Students and Teachers Calculus Problems - Alvirne High School Problems of the Week site The Cereal Box Problem by George Reese A Collection of Math-Olympiad-Problems by Hans Vernaeve Competitions Canadian Mathematical Society (CMS) Countries with Mathematical Olympiad Websites - Links to Mathematical Olympiad Websites. CRUX Mathematicorium Online The Digital Supplement to the Canadian Mathematical Society's Problem Solving Journal Ken Duisenberg's Puzzle of the Week Elementary Problem of the Week by Ruth Carver (MathForum) Fermat's Last Theorem Flanders Mathematics Olympiad The Goal Angle Problem Hilbert ... 's Tenth Problem by Karlis Podnieks The Mathematical Problems of David Hilbert by David E. Joyce

50. Numcom08 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 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.

51. Editing And Debugging M-Files (Development Environment) Debugging Example The collatz problem. The example debugging session The collatz problem is to prove that the Collatz function will resolve to 1 for http://www.phys.ufl.edu/docs/matlab/techdoc/matlab_env/edit_15a.html

Development Environment Debugging Example - The 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 for the next integer in the sequence 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 n and plots the results. The plot shows patterns that can be further studied. Following are the results when n is 1, 2, or 3.

52. Math Games: Gaussian Numbers The collatz problem has been tested out to 262•250 by the 3x+1 search. Apollonian Gasket , Catalan Conjecture , collatz problem , Complex Plane , http://www.maa.org/editorial/mathgames/mathgames_03_15_04.html

Search MAA Online MAA Home Math Games Gaussian Numbers Ed Pegg Jr., March 15, 2004 I learned about the complex plane at about the same time I learned of the quadratic formula . With that, I knew x^2 - 2 x + 17 had a solution in complex numbers. I didn't come across complex numbers again until I saw the Mandelbrot set for the first time. Figure 1. The Mandelbrot Set, colored by average differences between iterations. After that, I was fascinated by the complex plane. I remember a fellow enthusiast showing me the basic Mandelbrot set. "I managed to get it to render in just 8 hours!" Computers have sped up tremendously since then. There are dozens of free programs for Mandelbrot set explorations, allowing real-time zooms into all manner of beautiful structures. I can particularly recommend Fractal Extreme and Fractint Outside of fractals, one rarely sees complex numbers in the newspapers. For example, the number 2 - 1 recently made headlines as the new largest prime number, as part of the Great Internet Mersenne Prime Search +1, found by the

53. Nafindix - Mathematics The 3x+1 Problem is also known as the collatz problem. So the sequence of numbers in the collatz problem is generated by a repeated composition of the http://www.nafindix.com/math/math.html

Mathematics For my publications and preprints, see my academic website Below I have described some of my first discoveries in mathematics. These are the things that inspired me to become a mathematician. Symmetric Functions and the Binomial Theorem The Binomial Theorem The binomial theorem gives the following expansion: The Binomial Product Formula I have discovered a generalization of the binomial theorem that gives the expansion of a binomial product. The formula involves two operators: The first operator takes a function and returns a number, whereas the second operator takes a function and returns another function (like, for example, the differential operator). Using these operators, we can provide the following generalization of the binomial theorem: This is a paper I wrote describing this and related discoveries: Factorizations of Symmetric Function Relations  Ambiguity Division In examining a table of fractions, we readily observe that many of its elements are repeated. These are the fractions whose numerator and denominator have a common factor, allowing them to be reduced. In the figure below, all repeated fractions have been highlighted: This is a wider view: It turns out that the reciprocal of the portion of fractions that is repeated in this way (represented by the ratio of white to black above) is These are two informal papers I wrote detailing my analysis of this relationship: Ambiguity in a Mapping from Positive Integer Pairs to Rational Numbers - Nov. 2002

54. Collatz Conjecture - Definition Of Collatz Conjecture In Encyclopedia The Collatz conjecture, also known as the 3n + 1 conjecture, collatz problem Wolfram s MathWorld gives a good insight for introduction of this problem http://encyclopedia.laborlawtalk.com/Collatz_conjecture

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law The Collatz conjecture , also known as the n + 1 conjecture , the Ulam conjecture or the Hailstone sequence or Hailstone numbers , was first stated in 1937 and concerns the following process: Pick any positive integer n If n is even, divide it by two; if it is odd, multiply it by three and add one. If n = 1, stop; else go back to step 2. Or, algebriacally, f n n if f n n if (see modular arithmetic ). For instance, starting with n = 6, we get the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1. The secuence of Collatz conjecture can be easily computated by: The following is wikicode , a proposed pseudocode for use in many articles: function collatz(n) print n; if n = 1 return else if n mod return else return One can easly see that every time that n is odd, in the next call, will be even, because when n is replaced by 2 n + 1 then 3 n + 1 is replaced by where z n The Collatz conjecture says that this process always stops, no matter what the start value.

55. Università Degli Studi Di Ferrara - Cerca in rete Kudlek/Radu Stefan Niculescu Some results on the collatz problem. 3283 Benedict Seifert On the arithmetic of cycles for the CollatzHasse http://felix.unife.it/Root/d-Mathematics/d-Number-theory/b-3x 1

LA PAGINA RICERCATA NON ESISTE O E' STATA RIMOSSA. L'indirizzo del portale dell'Università degli Studi di Ferrara è : http://www.unife.it/ Aiuto In Internet unife.it

56. Mathematical Talks International Conference on the collatz problem and Related Topics. August 5, 1999 Katholische Universität Eichstät, Eichstät, Germany http://math.scranton.edu/monks/talks.html

Ken Monks Dept. of Mathematics University of Scranton Scranton, PA 18510 SITE CONTENTS Home Page Software Courses Publications ... Misc Phone: (570) 941-6101 Fax: (570) 941-5981 Office: STT163-A Email: monks@scranton.edu Mathematical Talks The purpose of this page is to make available the slides, preprints, images and other materials mentioned in some of my mathematical talks for the convenience of the audience members. Lehigh Mathematics Department Colloquium Jan 26, 2004 - Lehigh University, Bethlehem, Pa, USA My slide from the talk Links to articles and sites mentioned in the talk: publications by Scranton students my Publications Lehigh Mathematics Department Colloquium March 19, 2003 - Lehigh University, Bethlehem, Pa, USA My slides from the talk (in pdf format) Links to articles and sites mentioned in the talk: Eric Roosendaal's page publications by Scranton students my Publications ... attendees at the International Collatz Conference (see below) my Maple software for the Collatz problem International Conference on the Collatz Problem and Related Topics August 5, 1999 - Katholische Universität Eichstät, Eichstät, Germany

57. Nancy Street - Hobbies - Mathematica - Hailstone Numbers The study of Hailstone Numbers (often called The collatz problem) is not really a mathematical hobby of mine. I spent one morning casually playing with them http://www.orthogonal.com.au/hobby/mm/hail/

Hailstone Numbers Home Hobbies Mathematica Hailstone Numbers Preamble The study of Hailstone Numbers (often called The Collatz Problem ) is not really a mathematical hobby of mine. I spent one morning casually playing with them in Mathematica and I decided the results were worth saving on a web page in case anyone might be interested. Read on... Introduction On Sunday morning the 25th of June 1998 I was woken at 6am by the scratching sounds of a cat trying to bury a pool of it's stinking number ones on a rug in the Computer Room . After half an hour of cleaning I knew there was no point in trying to go back to sleep, so I doodled on my PC for a while trying to think of something to do. For some unknown reason I began to think about the old Hailstone Numbers from my school days. Calculating these number sequences became a bit of a fad (like Conway's game of Life ) back in the early computer days. I think it was the early 1970s, long before personal computers were conceived. I remember reading somewhere at the time that calculating the sequences wasted so much computer time that they were regarded as a communist plot to undermine computer research in the western world. As a reminder, Hailstone Numbers go like this:

58. Introduction To Artificial Intelligence Selmer Bringsjord Collatz Problem as demo of loop form; Click here to see code that produces A picture of the collatz problem More on the collatz problem and Related 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).

59. Prof. Dr. Manfred Kudlek Some Considerations on the collatz problem (AI, 37(2), p 145160, 2000) (Proceedings of International Conference on the collatz problem and Related http://www.informatik.uni-hamburg.de/TGI/publikationen/public/biblio_kudlek.html

Arbeitsbereich T HEORETISCHE G RUNDLAGEN DER I NFORMATIK Prof. Dr. Manfred Kudlek This page is also available in English. Cette page existe aussi en Bibliographie Publikationen (Diplomarbeit, DESY-Sonderdruck, Mai 1966, 33 p.) (Dissertation, DESY-Bericht 70/29, Juli 1970, 35 p.) (mit Erich H. Mickler) Solar and Lunar Eclipses of the Ancient Near East from 3000 B.C. to with Maps (Alter Orient und Altes Testament - Sonderreihe, Band 1, Butzon und Bercker, Kevelaer, 1971, 200 p.) A Statistical Analysis of Dates on Maya Monuments to Find Astronomical Inscriptions (Atti del XL. Congresso Internazionale degli Americanisti, Vol 1, p 431-432, Tilgher, Genova, 1973) (Atti del XL. Congresso Internazionale degli Americanisti, Vol 3, p 45-53, Tilgher, Genova, 1975) Characterization of Derivation Sets of Formal Systems (1. GI-Fachtagung Automatentheorie und Formale Sprachen, LNCS 2 , p 156-165, Springer, 1973) (Indiana 7, p 57-79, Gebr. Mann, Berlin, 1982) Comparing Several Ways of Context-independent Parallel Rewriting (4. Jahrestagung der GI, LNCS 26, p 122-130, Springer, 1975)

60. The Collatz Problem, Also Known As The 3x+1 Problem Ilan Vardi The The collatz problem, Also Known as The 3x+1 Problem Ilan Vardi The Collatz map is taken to be x x/2 if x is even and x - (3x+1)/2 if x is odd. http://www.cs.ucla.edu/~klinger/col.html

The Collatz problem, Also Known as The 3x+1 Problem Ilan Vardi The Collatz map is taken to be x -> x/2 if x is even and x -> (3x+1)/2 if x is odd. ... I. Vardi, Computational Recreactions in Mathematica, Addison-Wesley 1991, Chapter 7 ... the 4 known cycles.... Discussion: 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 4 known cycles (the program runs on positive and negative integers): An efficient algorithm is used to compute how many iterations there are up to a cycle (the total stopping time). This algorithm is discussed in detail in Computational Recreations in Mathematica, Chapter 7. BeginPackage["Examples`Collatz`"] <= Abs[n]

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