81. The 3x + 1 Problem And Its Generalizations A survey article by Jeff Lagarias. http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html

Contents Next: Introduction The Problem and its Generalizations Jeffrey C. Lagarias Murray Hill, NJ 07974 (January 16, 1996) A list of math activations A linked list of keywords ... Contents Next: Introduction

82. Collatz Problem -- From MathWorld 3x+1.htmlJeffrey 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.astro.virginia.edu/~eww6n/math/CollatzProblem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Mathematical Problems Prize Problems ... Martinez 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 required for 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

83. Welcome To The 3n + 1 Problem! http://math.carleton.ca/~amingare/mathzone/3n 1.html

84. Collatz Conjecture - Encyclopedia Article About Collatz Conjecture. Free Access, Click the link for more information. said about the collatz conjecture Mathematicsis not yet ready for such problems. He offered $500 for its solution. http://encyclopedia.thefreedictionary.com/Collatz conjecture

Dictionaries: General Computing Medical Legal Encyclopedia Collatz conjecture Word: Word Starts with Ends with Definition The Collatz conjecture , also known as the n +1 conjecture , the Ulam conjecture or the Hailstone sequence , was first stated around 1950 and concerns the following process: # Pick any positive integer The integers consist of the natural numbers (0, 1, 2, ...) and their negatives (-1, -2, -3, ...; -0 is equal to and therefore not included as a separate integer). The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which stands for Zahlen (German for "numbers"). They are also known as the whole numbers , although that term is also used to refer only to the positive integers (with or without zero). Click the link for more information. 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. For instance, starting with n = 6, we get the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1. The Collatz conjecture says that this process always stops, no matter what the start value.

85. Unsolved Problem 25 18Jun-1995 Unsolved problem 25 Start with any positive integer. Each week, foryour edification, we publish a well-known unsolved mathematics problem. http://cage.rug.ac.be/~hvernaev/problems/Proble25.html

18-Jun-1995 Unsolved Problem 25: Start with any positive integer. Halve it if it is even; triple it and add 1 if it is odd. If you keep repeating this procedure, must you eventually reach the number 1? For example, starting with the number 6, we get: 6, 3, 10, 5, 16, 8, 4, 2, 1. Reference: [Gardner 1983] Martin Gardner, Wheels, Life, and Other Mathematical Amusements. W. H. Freeman. New York: 1983. Page 196. Each week, for your edification, we publish a well-known unsolved mathematics problem. These postings are intended to inform you of some of the difficult, yet interesting, problems that mathematicians are investigating. We do not suggest that you tackle these problems, since mathematicians have been unsuccessfully working on these problems for many years. general references previous problem problem archive next problem ... electronic publications

86. Nancy Street - Hobbies - Mathematica - Hailstone Numbers Preamble. The study of Hailstone Numbers (often called The CollatzProblem) is not really a mathematical hobby of mine. I spent one 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 awoken at 6am by the scratching sounds of a cat trying to bury a pool of it's stinking pee 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:

87. Puzzle 213.  Hailstone Champion Sequences Hailstone sequences of numbers, {a i } are these produced in the socalled Collatzproblem for a given starting number a 1 (see 1 2), applying recursively 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?

88. Lothar F Mackert - ResearchIndex Document Query Daniel www.math.muluebeck.de/workers/potts/paper/illvorkon.ps.Z The 3n+1 CollatzProblem and Functional Equations - Berg, Meinardus (1995) (Correct) (1995 http://citeseer.ist.psu.edu/cis?q=Lothar F. Mackert

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