41. List Of Publications W. Thomas, World Scientific, Singapore (2000). A8 Andrei, St., Kudlek, M.,Niculescu, R., S. “Some Results on the collatz problem”, Proceedings of http://www.infoiasi.ro/~stefan/Lucrari.html

List of publications The articles published in international recenzed journals: Andrei, St.: "The Determinant of the Boolean Formulae ", Analele Stiintifice ale Universitatii Bucuresti, Informatica, Ano.XLIV, Bucuresti, Romania, pp. 83-92 (1995) Andrei, St.: "A Boolean Model for the Hadamard Matrix and the Walsh System", Journal of North China University of Technology , Vol. 7, Beijing, China, pp. 6-17 (1995) - Andrei, St.: "Finding Keys in Particular Classes of Functional Dependencies ", Analele Stiintifice ale Universitatii Al.I.Cuza, Informatica, Tomul IV, Iasi, Romania, pp. 5-12 (1995) Andrei, St., Kudlek, M.: "Morphological Grammars ", Analele Stiintifice ale Universitatii Al.I.Cuza, Informatica, Tomul V-VI, Iasi, Romania, pp. 85-113 (1997) Andrei, St., Masalagiu, C.: "About the Collatz Conjecture", Acta Informatica , No. 35, Vol.2, pp. 167-179 (1998) Andrei, St., Grigoras, Gh., Masalagiu, C., Rotaru, T.: “An efficient parallel algorithm for converting regular expressions into right linear grammars”, Analele Stiintifice ale Facultatii de Matematica, Informatica

42. Www.faqs.org/ftp/faqs/sci-math-faq/unsolvedproblems * collatz problem * Goldbach's conjecture * Twin primes conjecture _Names of large numbers http://www.faqs.org/ftp/faqs/sci-math-faq/unsolvedproblems

Newsgroups: sci.math,sci.answers,news.answers Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!spool.mu.edu!torn!watserv3.uwaterloo.ca!undergrad.math.uwaterloo.ca!neumann.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math FAQ: Unsolved Problems Summary: Part 18 of many, New version, Originator: alopez-o@neumann.uwaterloo.ca Message-ID:

43. Miller's Mathematical Ideas, 9th Edition Web Site Chapter 5 -- Internet Project the 1930s when it was studied by the German mathematician Lothar Collatz.For this reason it is often called the collatz problem. http://occawlonline.pearsoned.com/bookbind/pubbooks/miller2_awl/chapter5/essay1/

Chapter 5 Two Topics From Number Theory Back to List Introduction Perfect Numbers The 3n+1 Problem Introduction The two topics discussed here, perfect numbers and the 3 n +1 problem, have nothing obvious in common except for the following characteristics. Each topic: is very easy to describe, has been around for a fairly long time, stymies mathematicians to this day. These are classic examples of simple subjects for which the key result is still not known to mathematicians. Perfect Numbers You know what it means for one whole number to be a divisor of another. For example, 3 is a divisor of 12 since however 5 is not a divisor of 12 since 5 does not go into 12 a whole number of times. The number 1 is a divisor of every whole number and every number is a divisor of itself. Given any number, we can (with a little work) list all of its divisors. For example, a few numbers and their list of divisors appear in the table below. Number Divisors By looking at these few examples, you see a number may have just a few divisors or many divisors. A number such as 97 that has only two divisors, itself and 1, is a prime number Next we take the list of divisors, drop the number itself from the list, that is, consider only those divisors less than the number, and add up these divisors .

44. SearchUK - Finds It Fast! collatz problem From Eric Weissten's World of Mathematics. Articlewith references and links. The collatz problem is a special case. http://www.searchuk.co.uk/Top/Science/Math/Number_Theory/Open_Problems/Collatz_P

Home Top Science Math > Collatz Problem ADULT SHOPPING FINANCE GAMBLING ... 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 The 3x+1 Problem Search Results - Maximum number tested (N): 91·2^50 = 102456891522678784.

45. Index Page Of HENs (the New Symmetrical Notation Systems) Seminar new Using HEN3, collatz problem is solved. Back to the Top of this Page Indexof this Page. 3. Application of the HEN systems. collatz problem is solved. http://www.crt.or.jp/~kokochi/HENindxe.htm

Top Page / Index Page Japanese Page The English Pages are under construction. If you are using a Web Browser, which can handle LAYERs (ex. Navigator 4.0 you can enjoy some pages that explain the Notation Systems by principle movement. Integer table was newly built Text-Book (in Japanese) of this Seminar was published WANTed Cooperators, translating from Japanese to any other Languages. A Digital Balance of system is published. new Using HEN3, Collatz Problem is solved. Back to the Top of this Page Index of this Page 1. Comparison of some Notation systems Notation system using a digital Balance Explanatory notes of digital Balance some Notation systems the Decimal system : Radix = 10 the New Notation systems : Radix = 2, 3, 5, 7 Comparison of same Radix Notation systems Relation between a mixed decimal value and a position on the number line 2. Integer Table / HEN3 / HEN5 / HEN7 / DECimal 3. Application of the HEN systems Collatz Problem is solved. Back to the Index of this Page 1. Comparison of some Notation systems Notation system using a digital Balance Explanatory notes of digital Balance Back to the Index of this Page some Notation systems the Decimal system Arabic numeral system Roman numeral system Japanese abacus (Yotsu-dama abacus) ... Positive Radix Notation systems need minus (or plus) sign.

46. Felix.unife.it/Root/d-Mathematics/d-Number-theory/b-3x+1 Eichsta''t Conference 1999, 5p. 13469 Stefan Andrei/Manfred Kudlek/Radu Stefan NiculescuSome results on the collatz problem. Eichsta''t Conference 1999, 15p. http://felix.unife.it/Root/d-Mathematics/d-Number-theory/b-3x 1

47. Practical Foundations Of Mathematics function pList({ a,b})\rightharpoonup N with p( ) = 1, p(cons(a, l)) = a(p(l))and p(cons(b,l)) = b(p(l)). The collatz problem asks whether p is surjective. http://www.dcs.qmul.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).

48. Problème De Syracuse Translate this page Il a pris de nombreux noms depuis (collatz problem, conjecture de Thwaites, algorithmede Hasse, ) suite aux travaux de chercheurs américains intrigués http://www.sciences-en-ligne.com/momo/chronomath/anx2/pb_syracuse.html

Problème de Syracuse Ce "petit" problème d'arithmétique apparaît pour la première fois, en 1937, aux Etats-Unis, posé par le professeur Lothan Collatz de l'université de Syracuse (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; sil est impair, on le triple et on ajoute 1; on itère le procédé sur le nouvel entier obtenu. 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... Le problème de Syracuse est un problème ouvert : des études très poussées n'ont pas, à ce jour (8 février 2002), conduit à la preuve de cette conjecture.

49. Mathematical Problems - Problem Solving - Mathematical Competitions (Math Links Mathematics Hots (Problems) by Bruno KeviusCategory Science Math Recreations...... The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Kakutani'sproblem, Hasse's algorithm, and Ulam's problem by Jeff Lagarias; http://www.abc.se/~m9847/matre/problem.html

Mathematical Resources Math Links by Bruno Kevius This list is continually under development 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) 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 ALVIRNE HIGH SCHOOL PROBLEM OF THE WEEK SITE Andrei Jorza's Home Page Calculus Problems Alvirne High School in Hudson, N.H. The Cereal Box Problem by George Reese A Collection of Math-Olympiad-Problems by Hans Vernaeve 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 Paul Erdos: An I nfinity of Problems - Ivars Peterson (MathLand) International Mathematical Olympiad / Olympiade internationale ... International Mathematics Olympiad (IMO) Eindhoven University of Technology

50. 3x+1 Conjecture Verification Results Maximum number tested (N) 91·2^50 = 102456891522678784.Category Science Math Number Theory Open Problems collatz problem...... The 3x+1 conjecture 1, 2, problem E16 asserts that starting from any positiveinteger n the repeated iteration of T(x) eventually produces the integer 1 http://www.ieeta.pt/~tos/3x 1.html

51. Editing And Debugging M-Files (Development Environment) the use of the Editor/Debugger and debugging function using an example.Closing MFiles, Debugging Example - The collatz problem, http://www-h.eng.cam.ac.uk/help/tpl/programs/help/techdoc/matlab_env/edit_14a.ht

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 debugging functions using a simple example. It includes these topics: Types of Errors Finding Errors Debugging Example - The Collatz Problem Trial Run for Example ... Using Debugging Features Types of Errors Debugging is the process by which you isolate and fix problems with your code. Debugging helps to correct two kinds of errors: Syntax errors - For example, misspelling a function name or omitting a parenthesis. MATLAB detects most syntax errors and displays an error message in the Command Window describing the error and showing its line number in the M-file. Run-time errors - These errors are usually algorithmic in nature. For example, you might modify the wrong variable or perform a calculation incorrectly. Run-time errors are apparent when an M-file produces unexpected results. Finding Errors Usually, it's easy to find syntax errors based on MATLAB's error messages. Run-time errors are more difficult to track down because the function's local workspace is lost when the error forces a return to the MATLAB base workspace. Use the following techniques to isolate the cause of run-time errors: Remove selected semicolons from the statements in your M-file. Semicolons suppress the display of intermediate calculations in the M-file. By removing the semicolons, you instruct MATLAB to display these results on your screen as the M-file executes.

52. Index Of Examples Development Environment. Editing and Debugging MFiles. Debugging Example- The collatz problem. Improving M-File Performance - The Profiler. http://www-h.eng.cam.ac.uk/help/tpl/programs/help/techdoc/demo_example.html

Index of Examples Use this index to link directly to examples in the documentation. Development Environment Editing and Debugging M-Files Debugging Example - The Collatz Problem Improving M-File Performance - The Profiler An Example Using the Profiler Mathematics Polynomials and Interpolation Multidimensional Data Gridding Triangulation and Interpolation of Scattered Data Data Analysis and Statistics Analyzing Residuals Polynomial Fit Exponential Fit Error Bounds ... Using the Basic Fitting Interface Function Functions Minimizing Functions of One Variable Plotting Mathematical Functions Computing the Length of a Curve Differential Equations - Initial Value Problems ODE Initial Value Problem Examples Solving an IVP ODE in MATLAB (van der Pol Equation, Nonstiff) Simple Nonstiff Problem Stiff Problem (van der Pol Equation) ... Differential-Algebraic Problem Differential Equations - Boundary Value Problems ODE Boundary Value Problem Examples Mathieu's Equation Using Continuation to Solve a Difficult BVP Partial Differential Equations PDE Examples A Single PDE Electrodynamics Problem Sparse Matrices Generating a Second Difference Operator Programming and Data Types M-File Programming Simple Function Example Simple Script Example Passing Variable Numbers of Arguments Checking the Number of Function Arguments ... Conversions Between Date Formats Character Arrays (Strings) Comparing Strings For Equality Converting to a Cell Array of Strings Converting Characters to Numeric Values Creating Two-Dimensional Character Arrays ... Categorizing Characters Within a String Multidimensional Arrays

53. Untitled The collatz problem, Also Known as The 3x+1 Problem Ilan Vardi The Collatz mapis 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]

54. Links To Other Mathematical Recreations, Games And Puzzles The hailstone problem, aka The 3x+1 problem, collatz problem, the Syracuseproblem, Kakutani's problem, Hasse's algorithm, and Ulam's problem. http://bruichladdich.dcs.st-and.ac.uk/mathrecsFolder/links.html

55. Mathsoft: Mathsoft Unsolved Problems: Unsolved Problems On Other Sites also Keith Matthew's The Generalized 3x+1 Mapping (University of Queensland) and1999 Conference on the collatz problem Proceedings (Eichstätt, Germany); http://www.mathsoft.com/mathresources/problems/article/0,,1999,00.html

search site map about us  + news  + ... Unsolved Problems Unsolved Problems Links On a Generalized Fermat-Wiles Equation Zero Divisor Structure in Real Algebras Sleeping Habits of Armadillos Engineering Standards ... Math Resources Unsolved Problems on Other Sites Jeff Lagarias' 3x+1 problem and related problems The Generalized 3x+1 Mapping (University of Queensland) and 1999 Conference on the Collatz Problem Proceedings Alex Lopez-Ortiz's sci.math FAQ on Famous Problems in Mathematics (University of New Brunswick) Chris Caldwell's Riemann Hypothesis (University of Tennessee at Martin); also Daniel Bump's Riemann Hypothesis (Stanford University) and Barry Cipra's A Prime Case of Chaos MathPro Press Unsolved Math Problem of the Week column and General References Twin Primes Conjecture and Goldbach's Conjecture, discussed in Prime Numbers: Some Unsolved Problems , part of MacTutor History of Mathematics (University of St Andrews); also Chris Caldwell's Prime Conjectures and Open Questions (University of Tennessee at Martin) Jan Otto Munch Pedersen's Known Amicable Pairs (Vejle Business College, Denmark) and Chris Caldwell's

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

Busy Beaver Currently Known Results The function Sigma (n) denotes the maximal number of tape marks which a 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 Local Resources (English) The author's paper Attacking the Busy Beaver 5 , Bulletin of the EATCS, Number 40, February 1990, pp. 247-251, [ISSN 0252-9742]

57. Editing And Debugging M-Files (Development Environment) the use of the Editor/Debugger and debugging functions using an example.Closing MFiles, Debugging ExampleThe collatz problem, http://ftp.math.hkbu.edu.hk/help/techdoc/matlab_env/edit_d20.html

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

58. Untitled Conway has shown that a generalized version of the collatz problem, using remaindersmodulo numbers larger than 2, is universal (meaning that it can simulate http://cs.stmarys.ca/~dawson/abacus.html

A two-register abacus applet This is a Java applet that demonstates the operation of a two-register abacus. This is about the ultimate in RISC computing - a computer with two registers, and five operations: INCX, INCY, DECXJMP, DECYJMP, and HALT. It was shown by Minsky in 1967 that despite its simplicity this class of machines contains a machine that is "universal" - with the proper program (coded into the contents of its two registers!) it can simulate any computer whatsoever. The INC* commands just increment one register. The DEC*JMP commands try to decrement, but if the contents of the register is they branch to a different instruction instead . The HALT command just causes the machine to sit there and look pretty. The universal machine is a rather opaque hack involving several levels of simulation, and you cannot really see how it works at a glance. What I've exhibited here is a two-register abacus hard-wired to do the Collatz "3n+1" problem. Click on the big green button to start! In this famous problem, you start with a number (a starting point of 7 is hardcoded into this version of the program; I hope that later versions will let you choose your own favorite number and do other cool things.) If the number's even you divide by 2; if it's odd you multiply by 3 and add 1. Then repeat until either you loop or you get to 1.

59. Hw2 John Conway proved that the original collatz problem has no nontrivialcycles of length less than 400. Lagarias (1985) showed that http://www.csee.umbc.edu/331/fall02/0101/homework/hw6/

CMSC331 Programming Languages, section 0101, Fall 2002 Homework Hailstone Sequences in Java out: , due: This assignment asks you to write a very simple program in Java. It is intended to give you some experience in writing and debugging a simple Java program. Consider the following problem: Choose a positive integer and repeatedly do the following: if the number is 1, quit; if the number is even, cut it in half; and if the number is odd, multiply it by 3 and add 1. For example, if you start with the number 17, you get the sequence: . Generate the sequence from a starting number and you'll find the numbers go up and down like a hailstone in a cloud before it plummets to earth (e.g., one). Background Does this procedure eventually reach 1 and stop for every choice of starting number? So far, this is an unsolved problem no one has yet proved that the process always results in 1, and no one has yet found a counterexample. This problem was first posed by L. Collatz in 1937 and goes under several names: the Collatz conjecture, the ' n +1 conjecture ', the Ulam conjecture, and the Syracuse problem. The sequence is also commonly called the

60. Index Of Examples Editing and Debugging MFiles Debugging Example - The collatz problem ImprovingM-File Performance - The Profiler An Example Using the Profiler. Mathematics. http://www.mathworks.ch/access/helpdesk_r12p1/help/techdoc/demo_example.shtml

Documentation MATLAB Index of Examples Use this index to find examples in the documentation. Select the a MATLAB topic area from the following list: Development Environment Mathematics Programming and Data Types Graphics ... External Interfaces Development Environment Editing and Debugging M-Files Debugging Example - The Collatz Problem Improving M-File Performance - The Profiler An Example Using the Profiler Mathematics Polynomials and Interpolation Multidimensional Data Gridding Triangulation and Interpolation of Scattered Data Data Analysis and Statistics Analyzing Residuals Polynomial Fit Exponential Fit Error Bounds ... Using FFT to Calculate Sunspot Periodicity Function Functions Minimizing Functions of One Variable Plotting Mathematical Functions Computing the Length of a Curve Differential Equations - Initial Value Problems ODE Initial Value Problem Examples Solving an IVP ODE in MATLAB (van der Pol Equation, Nonstiff) Simple Nonstiff Problem Stiff Problem (van der Pol Equation) ... Differential-Algebraic Problem Differential Equations - Boundary Value Problems ODE Boundary Value Problem Examples Mathieu's Equation Using Continuation to Solve a Difficult BVP Using Continuation to Verify a Solution's Consistent Behavior Partial Differential Equations PDE Examples A Single PDE Electrodynamics Problem Sparse Matrices Generating a Second Difference Operator Programming and Data Types M-File Programming Simple Function Example Simple Script Example Passing Variable Numbers of Arguments Checking the Number of Function Arguments ... Conversions Between Date Formats Character Arrays (Strings)

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