61. Kaprekar Number The 3n+1 Problem (collatz problem) top Take any natural number, from which you derive a collatz problem (3n+1Problem). Math Fun Facts (Francis Su) http://www.mathematische-basteleien.de/kaprekar.htm

The Kaprekar Number and more playings with numbers Contents of this Page What is the Kaprekar Number? The 3n+1 Problem The Steinhaus Cyclus The 196-Problem The Number 1089 ... To the Main Page "Mathematische Basteleien" What is the Kaprekar Number? The number 6174 is called the Kaprekar number. The Indian mathematician D.R.Kaprekar made the following discovery in 1949. (1) Take a four-digit number with different digits (acbd with (2) Form the largest and the smallest number from these four digits (dcba and abcd). (3) Find the difference of these digits. Maybe this is 6174 (dcba - abcd = 6174?). If it is not, form the largest and the smallest number from the difference and subtract these numbers again. You may have to repeat this procedure. The end result is always 6174, but there are no more than 7 steps. 1st example: Take the number 1746. 1st step: 7641 - 1467 = 2nd example: Take the number 5644. 1st step: 6544 - 4456 = 2088 2nd step: 8820 - 0288 = 8532 3rd step: 8532 - 2358 = 3rd example: Take the number 7652. 1st step: 7652 - 2567 = 5085 2nd step: 8550 - 0558 = 7992 3rd step: 9972 - 2799 = 7173 4th step: 7731 - 1377 = 6354 5th step: 6543 - 3456 = 3087 6th step: 8730 - 0378 = 8352 7th step: 8532 - 2358 = The problem is solved. (Spektrum der Wissenschaft, Erstausgabe 1978)

62. Collatz Conjecture - Wikipedia, The Free Encyclopedia Paul Erdos said about the Collatz conjecture Mathematics is not yet ready for such problems. collatz problem on MathWorld http://en.wikipedia.org/wiki/Collatz_conjecture

Collatz conjecture From Wikipedia, the free encyclopedia. The Collatz conjecture is an unresolved conjecture in mathematics . It is named after Lothar Collatz , who first proposed it in . The conjecture is also known as the n , the Ulam conjecture (after Stanislaw Ulam ), the Syracuse problem , as the hailstone sequence or hailstone numbers , or as Wondrous numbers as per G¶del, Escher, Bach . It asks whether a certain kind of number sequence always ends in the same way, regardless of the starting number. Paul Erdős said about the Collatz conjecture: "Mathematics is not yet ready for such problems." He offered $500 for its solution. Contents Statement of the problem Examples Program to calculate Collatz sequences Supporting arguments ... edit Statement of the problem Consider the following operation on an arbitrary positive integer If the number is even, divide it by two. If the number is odd, triple it and add one. For example, if this operation is performed on 3, the result is 10; if it is performed on 28, the result is 14. In mathematical notation, define the

63. Zeal.com Provides a visualization of the collatz problem using the Fractint iterated Provides a collection of links to resources about the collatz problem http://www.zeal.com/users/contributed_sites.jhtml?uid=10000

64. Alvaro Lozano S Homepage The collatz problem (or 3n+1 Problem). Some websites about the problem Lagarias, Also known as the Collatz Conjecture, Ulam s Problem, 14-2 Problem, http://www.colby.edu/personal/a/alozano/ma122/ma122.html

65. Puzzle 213.  Hailstone Champion Sequences According to the rules of the collatz problem for generating the hailstone sequences, M(a1)=a1 only if a1=2^k, for k= 2; otherwise, M(a1) a1. http://www.primepuzzles.net/puzzles/puzz_213.htm

Puzzles Puzzle 213. Hailstone Champion Sequences Hailstone sequences i Collatz problem for a given starting number a (see ), applying recursively the following rule: a i+1 = 3*a i +1 if a i is odd; otherwise a i+1 = a i For each particular initial value a there is only one maximal member, M(a ) in the corresponding sequence. For example if a =7 , the hailstone sequences is: and M(7)=52 According to the rules of the Collatz problem for generating the hailstone sequences, M( a a only if a a a The quotient Q( a )=M( a a is a measure of the height of M( a ) relative to the initial value a of the corresponding hailstone sequence. I have produced a table of " champion hailstone sequences ", for a M( a Q( a a =1; then we annotate in the next row, the next earliest a value such that its Q(a ) is larger than the Q(a ) of the previous row; and so on... a M( a Q( a Questions: Does Q( a ) grow beyond any limit? 2. Do you devise any special (non-trivial) property for the a values in the table of champions hailstone sequences?

66. Puzzle 73.- A Collatz-like Sequence. (A Puzzle Suggested By Jack Brennen ) This is obviously derived from the collatz problem, and in fact, all even or prime numbers = 3 have the same successor as they do in the Collatz sequence; http://www.primepuzzles.net/puzzles/puzz_073.htm

Puzzles Puzzle 73.- A Collatz-like sequence. (A puzzle suggested by Jack Brennen Consider the sequence of integers B[] defined by a starting point B[0] and the rules: If B[n] is prime, B[n+1] is defined as 3*B[n]+1. If B[n] is composite, B[n+1] is defined as the largest proper divisor of B[n], which can also be described as B[n]/P, where P is the smallest prime factor of B[n]. The question is this : Does every starting point eventually end up entering the cycle The same question can be posed with the rule changed so that prime numbers go to 3*B[n]-1 instead of 3*B[n]+1. Do all starting points then eventually enter this cycle? Solution Felice Russo got the following interesting results: " Question 1 . I have checked (with an Ubasic program) that all the integers between and terminate in the cycle: 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2. Moreover for the first 1000 integers I calculated the number of steps needed to end up in the cycle. I indicate this quantity with T(n) and we can call it the "transient time". Then I calculated the ratio T(n)/n ("Normalized Transient time") which behavior is reported in the below graph.

67. [Collatz 3n+1 Conjecture And NKS] - A New Kind Of Science The NKS GM/0312309 which proves that the 3n+1 problem (Collatz conjecture) is I m claiming that the collatz problem (and probably RH too) can never be settled http://forum.wolframscience.com/archive/topic/606-1.html

A New Kind of Science: The NKS Forum Pages: Collatz 3n+1 Conjecture and NKS (Click here to view the original thread with full colors/images) Posted by: Craig Feinstein This weekend, I found NKS in my local library, checked it out, and was pleasantly surprised - I haven't read it all yet but I saw that Wolfram introduces the notion of "computationally irreducible" and predicts that many unsolved problems in math are so because they are computationally irreducible. I find this prediction very plausible, as I recently wrote a paper http://arxiv.org/abs/math.GM/0312309 which proves that the 3n+1 problem (Collatz conjecture) is unprovable, using this idea. Any comments on my paper and how this relates to NKS are welcome. I look forward to reading more of NKS. Posted by: AnthonyMartin Craig, Your approach is interesting, but I have a few questions. Regarding notation in your paper: Also: Posted by: Craig Feinstein Anthony, thank you for reading my paper. If you or others have any other questions, please ask. Regarding notation in your paper: I meant that x is a string of m bits.

68. MATHEMATICAL INTERESTS, PAST And PRESENT I spent a considerable amount of time helped by collaborators, studying certain aspects of the 3x+1 problem (collatz problem) and related problems. http://www.numbertheory.org/interests.html

Mathematical Interests, Past and Present Early Years (1958-63) The first number theory book that I studied was L.E. Dickson's "Introduction to number theory". It was available as a cheap Dover book at the time. It must surely rank as the dullest book on number theory ever written. In 1960, I persuaded Prof. Clive Davis to replace a course on projective geometry which used to be given by H.K. Powell, by a course on number theory. This was the first time number theory had been taught here. Clive's course was very stimulating. I went on to study E. Landau's Primzahlverteilung and struggled through Atle Selberg's proof of the Prime Number Theorem and H.N. Shapiro's proof of Dirichlet's theorem, as well as other papers on elementary analytic number theory. My first discovery, in 1962, after perusing the relevant section in Hardy and Wright's book, was the prime-producing polynomials of Euler and their connection with imaginary quadratic fields with class number one. I soon learned that D.H. Lehmer had already obtained this result in Sphinx, Bruxelles, 1933. Harvey Cohn's A Second Course in Number Theory 1963, also had an account.

69. Heiner Marxen - Busy Beaver In einer Seite Prinzipielle Grenzen der Berechenbarkeit schreibt Arno Schwarz über Turing Maschinen, Busy Beaver und das collatz problem (3n+1). http://www.drb.insel.de/~heiner/BB/

Busy Beaver Chapter Table: Currently Known Results News / Recent Changes Local Resources (English) Other Resources (English) ... Other Resources (German/Deutsch) Currently Known Results The function Sigma (n) denotes the maximal number of tape marks which a 2-symbol Turing Machine (TM) with n internal states and a two-way infinite tape can produce onto an initially empty tape and then halt. The function S (n) denotes the maximal number of steps (shifts) which such a TM can do (it needs not produce many tape marks). The following table gives some known values: n Sigma (n) S (n) Source Lin and Rado Lin and Rado Lin and Rado Brady Marxen and Buntrock Marxen and Buntrock Note: The values for n=6 have been verified independently by Paul R. Stevens and by Clive Tooth. The exact numbers are found in the new list of 6-state record machines A general method due to Milton W. Green produces (computable, but not primitive recursive) lower bounds of Sigma for every n. He gives Sigma Of course, if you find an error in the above table, or can extend it... please let me know News / Recent Changes 10-Sep-2005: Added to the simulation results yet another 3x3 TM from Gregory Lafitte and Christophe Papazian, yielding amazing 1'525'688 nonzero symbols in over 9.8

70. Science: Mathematics: Number Theory: Open Problems - Open Site collatz problem. Also known as the 3n+1 problem, Syracuse problem, Thwaites problem. Define a function on positive integers n 3n+1 if n is odd, http://open-site.org/Science/Mathematics/Number_Theory/Open_Problems/

Open Site The Open Encyclopedia Project home submit content become an editor the entire directory only in Number_Theory/Open_Problems Top Science Mathematics Number Theory : Open Problems ABC Conjecture We define the radical R(n) of an integer n to be the the product of its distinct prime divisors. ABC Conjecture: For every k>1, there is a constant C k such that if a,b,c are coprime positive integers satisfying a+b=c, then c < C k R(abc) k The conjecture was proposed by Osterle and Masser in 1985. At present the best that can be proved is that c < exp(R(abc)^f) for a suitable f. Mason has proved the analogue of the conjecture for polynomials. If true, the conjecture would have numerous important consequences: among them would be another proof of Fermat's Last Theorem. We define the ABC ratio for a triple (a,b,c) to be A = log(c) / log(R(abc)). The conjecture implies that A is bounded, so it is of interest to find large values of A. The current best is: a=2, b=3 .109, c=23 giving A = 1.6299 Collatz Problem Also known as the 3n+1 problem, Syracuse problem, Thwaites problem. Define a function on positive integers n -> 3n+1 if n is odd, n/2 is n is even. Does every positive integer eventually end up in the cycle 4 -> 2 -> 1 -> 4?

71. Homepage For CSCI6420 - Spring 2005 The collatz problem Give a midlevel description of a Turing machine which determines whether an input binary number terminates under the Collatz iteration http://www.cs.ecu.edu/~hochberg/spring2005/csci6420.html

Grades Exams One midterm and one final, each worth 100 points. These may be canceled if the homeworks are going well enough. Homework (see below for assignments) Homeworks will be assigned from time to time. There will be 24 five-point problems assigned. You are allowed to redo any problems which you submitted by the due date. Arbitrarily many redos will be allowed. Final Grade If n tests are given, then your total T is equal to the value T X X n n + 1), where X is your homework grade, and X i is your grade on the i th exam. Your final grade is determined as follows: You will receive: at least an A if T at least a B if T at least a C if T 70 or less means an "F." Homework Assignments Please use the Discussion Board to ask questions about the homework. Click on Communication , then Discussion Board Problem set 1 Due January 31 (1.24 and 1.26) (1.27 and 1.28) (1.36 and 1.38) Problem set 2 Due February 21 1. Let M be a deterministic PDA with a stack whose size is limited to 3 elements. Give a context-free language which M cannot recognize, and prove that M cannot recognize it. 2. Build PDAs (you may use nondeterminism) to recognize each of the following languages:

72. Utah Logic: Misc: Hailstone (3n+1) Problem It is also frequently called the collatz problem. A google search on hailstone problem gives a lot of sites on the topic. http://www.utah.edu/utahlogic/misc/hailstone.html

Misc: Hailstone (3n+1) problem The hailstone problem has been a repeated discussion topic at the weekly meetings. (With new and interesting observations about it sometimes presented) So... probably there should be some explanation of what the problem is... If you start with a positive integer, multiply it by three, add one to the result, and then remove all factors of two, you will get some result. It is observed that if you do this repeatedly you will eventually hit the result ONE. There is, however, neither a proof that the sequence MUST hit 1, nor is there a counterexample known. (Although empirically it is known that there is no counterexample less than 10 billion) The problem has appeared in print numerous times (Apparently, for example, in the January 1984 issue of Scientific American). It is also frequently called the Collatz problem. A google search on "hailstone problem" gives a lot of sites on the topic. It is currently known (as reported by http://mathworld.wolfram.com/CollatzProblem.html , which I take as a trustable source) that there are no counter examples less than 10,000,000,000,000,000 Go to ...

73. Editing And Debugging M-Files (Development Environment) Debugging ExampleThe collatz problem Trial Run for Example Debugging is the process by which you isolate and fix problems with your code. http://www.rz.uni-freiburg.de/dienste/plattformen/ibm/compserver/frames/matlab/t

Development Environment Debugging M-Files This section introduces general techniques for finding errors, and then illustrates MATLAB debugger features found in the Editor/Debugger and equivalent debugging functions using a simple example. It includes these topics: Types of Errors Finding Errors Debugging ExampleThe Collatz Problem Trial Run for Example ... Using Debugging Features In addition to the Debugger and debugging functions, the Profiler included with MATLAB can be a useful tool to help you improve performance and detect problems in your M-files. For details, see Measuring Performance in the Programming and Data Types section of the MATLAB documentation. Types of Error s Debugging is the process by which you isolate and fix problems with your code. Debugging helps to correct two kinds of errors: Syntax errorsFor example, misspelling a function name or omitting a parenthesis. Syntax Highlighting helps you identify these problems, as does the process of setting breakpoints. When you run an M-file with a syntax error, MATLAB will most likely detect it and display an error message in the Command Window describing the error and showing its line number in the M-file. Click the underlined portion of the error message, or position the cursor within the message and press Ctrl+Enter . The offending M-file opens in the Editor, scrolled to the line containing the error. Use the

74. Print This Page: /mathsoft_resources/unsolved_problems/1999.asp 1999 Conference on the collatz problem Proceedings http//www.math.grin. edu/~chamberl/conf.html. 5. sci.math FAQ on Famous Problems in Mathematics http://www.mathsoft.com/mathsoft_resources/unsolved_problems/1999.asp?printUrl=/

75. IRMA Strasbourg - Publication 2000 Motsclé, Keywords collatz problem,iteration of arithmetical functions,exponential diophantine equations,udecidability. Classifications 11K38, 11K55, http://www-irma.u-strasbg.fr/irma/publications/2000/00018.shtml

Edouard Belaga,Maurice Mignotte Cyclic Structure of Dynamical Systems Associated with $3x+d$ Extensions of Collatz Problem. cliquez ici Collatz problem,iteration of arithmetical functions,exponential diophantine equations,udecidability Classifications : Contact : belaga@math.u-strasbg.fr Nature du texte, text type : Preprint Langue du texte : Anglais Maintained by Authors, May 2000 SXB 00018, Fri 26 May 00 Titres : Autres sites IRMA Strasbourg

76. LESSON PLANET - 30,000 Lessons And 1175 Web Sites For Problem Solving LESSON PLANETSearch Results for keyword Problem Solving. also known as the collatz problem, the Syracuse problem, Kakutani s problem, Hasse s algorithm http://www.lessonplanet.com/search/search?keywords=Problem Solving&display=websi

77. Behavior Of Busy Beavers Because of sameness with collatz problem, these problems are called Collatzlike problems. Thus, for each machine above (except for the runner-up with 6 http://www.logique.jussieu.fr/~michel/beh.html

Behavior of busy beavers Contents Turing machines with 5 states and 2 symbols Turing machines with 6 states and 2 symbols Turing machines with 3 states and 3 symbols Turing machines with 2 states and 4 symbols ... Collatz-like problems Last update : September 2005 Introduction How do good machines behave? We give below the tricks that allow them to reach high scores. A configuration of the Turing machine M is a description of the tape. The position of the tape head and the state are indicated by writing together between parentheses the state and the symbol currently read by the tape head. For example, the initial configuration on a blank tape is: You can see the introduction to Turing machines for more informations. We denote by a k the string a . . . a, k times. Turing machines with 5 states and 2 symbols Marxen and Buntrock's champion See also the simulation by Heiner Marxen This machine is the record holder in the Busy Beaver Competition for machines with 5 states and 2 symbols, since 1990. Marxen and Buntrock (1990) s(M) = 47,176,870 =? S(5,2)

78. EDUCATION PLANET - 1175 Web Sites For Problem Solving The 3x+1 problem and its generalizations * The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Kakutani s problem, http://www.educationplanet.com/search/Math/Problem_Solving

Search 100,000+ top educational sites, lessons and more! Home Math Found 1175 ' Problem Solving ' Web Sites. Also for ' Problem Solving 3424 Lesson Plans 59 Books 4 Software Titles 2 Maps ... 82 Other Products Web Sites (1 - 10 of 1175): 21st Century Problem Solving (cached) - Having trouble solving those word problems? This site will turn that around. Sound problem solving concepts are explained and examples are described that apply to grade school through graduate school and beyond in all subjects that use mathematics. Grades: Cache Report Link Problem Add Comment Algebra Story and Word Problems - 21st Century Problem Solving introduces how to solve problems reliably in Grade school through grad school and beyond in all subjects that use mathematics for problem solving Grades: Cache Report Link Problem Add Comment Algorithms and Ice Cream for All - Where shall we locate ice cream stands in our town so that no one has to travel too far to buy a treat? The problem-solving strategies for this problem apply to many other situations that require planning for facilities. Students will have a chance to gra... Grades: Cache Report Link Problem Add Comment Mathematics Problem Solving - Free Worksheets Grades: Cache Report Link Problem Add Comment EMT 525/725 Page - Department of Mathematics Education EMAT 4600/6600 Problem Solving in Mathematics. This is the web site page devoted to EMAT 4600/6600 Problem Solving in Mathematics, at the University of Georgia, as led by Jim Wilson.

79. Problème De Syracuse Translate this page Il a pris de nombreux noms depuis (collatz problem, conjecture de Thwaites, algorithme de Hasse, ), suite aux travaux de chercheurs américains intrigués http://serge.mehl.free.fr/anx/pb_syracuse.html

Problème (ou conjecture de Syracuse Ce "petit" problème d'arithmétique apparaît pour la première fois, en 1937, aux États-Unis, posé par le professeur Lothar Collatz (1910-1990) de l'université de Syracuse (non pas en Sicile mais dans l'État de New York, USA). Il a pris de nombreux noms depuis ( Collatz problem , conjecture de Thwaites, algorithme de Hasse , ...), suite aux travaux de chercheurs américains intrigués par ce problème. Le voici : On se donne un entier naturel n non nul : s'il est pair, on le divise par 2; s'il est impair, on le triple et on ajoute 1; on itère le procédé sur le nouvel entier obtenu tant qu'il est supérieur à 1. Dans tous les cas essayés depuis son origine, cet algorithme conduit à 1 (en finissant toujours par 4, 2, 1). 7 est le premier cas non trivial... : (mais si, mais si) Si n est une puissance de 2, la suite est évidemment strictement décroissante dans N et converge donc vers 1. Les autres cas sont moins évidents... ) ont été testés sur ordinateur : ça marche tout le temps...

80. Science/Math/Number_Theory/Open_Problems/Collatz_Problem The Collatz, Syracuse or 3x+1 problem. / Science / Math / Number_Theory / Open_Problems / collatz problem......Search. Category http://www.science-and-research.com/Science/Math/Number_Theory/Open_Problems/Col

Search: Category Description: The Collatz, Syracuse or 3x+1 problem Science Math Collatz Problem The Structure of the 3x + 1 Function Papers by Peter Schorer describing several new approaches. 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/ Collatz 3n+1 Problem Structure Observations posted by Ken Conrow to stimulate further research. URL: http://www-personal.ksu.edu/~kconrow/ Collatz Problem From Eric Weissten's World of Mathematics. Article with references and links. URL: http://mathworld.wolfram.com/CollatzProblem.html The 3x+1 Problem Conference Proceedings Katholische Universit¤t Eichst¤tt Germany; 56 August 1999. On-line proceedings and group photo.

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