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Collatz Problem: more detail

Conjectures: P Versus Np Problem, Poincaré Conjecture, Collatz Conjecture, Hilbert's Problems, Catalan's Conjecture, Sierpinski Number

Optimization Problems (Applied Mathematical Sciences) (Volume 0) by L. Collatz, W. Wetterling, 1975-07-23

Numerical Treatment of Free Boundary Value Problems: WORKSHOP NUMER.TREATMEnt Free Bound.Value Probl.Oberwolfach (International Series of Numerical Mathematics) by ALBRECHT, COLLATZ, et all 1982-06-14

Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations: CONFERENCE AT THE OBERwolfach Math.Res.Inst.,19.-25.11.1978 (International Series of Numerical Mathematics) by ALBRECHT, COLLATZ, et all 1979-01-01

Numerical Treatment of Eigenvalue Problems, Vol. 3 (International Series of Numerical Mathematics) by ALBRECHT, COLLATZ, et all 1984-05-31

NUMERICAL TREATMENT OF FREE BOUNDARY VALUE PROBLEMS by J.; Collatz, L.; et al; editors Albrecht, 1982

Numerical Treatment of Eigenvalue Problems: Workshop in Oberwafach, February 25-March 3, 1990/Numerische Behandlung Von Eigenwertaufgaben : Tagung (International Series of Numerical Mathematics) by J. Albrecht, Lothar Collatz, et all 1991-05

Numerical Treatment of Eigenvalue Problems (International Series of Numerical Mathematics)

OPTIMIZATION PROBLEMS.Translated by P.Wadsack.*(Applied Mathematical Sciences,17) by L./Wetterling,W. Collatz, 1975-01-01

The Dynamical System Generated by the 3n+1 Function (Lecture Notes in Mathematics) by Günther J. Wirsching, 1998-03-20

81. Citebase - Two Cellular Automata For The 3x+1 Map
G/A, 7 Guy, RK Collatz s Sequence. xE16 in Unsolved Problems in Number G/A, 16 EW Weisstein, collatz problem. From MathWorldA Wolfram Web
http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:nlin/0502061

82. Collatz_conjecture
http//mathworld.wolfram.com/CollatzProblem.html collatz problem on MathWorld deCollatzProblem frConjecture de Syracuse ko?
http://copernicus.subdomain.de/Collatz_conjecture
Suche:
Main Page
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 '''3''n'' + 1 conjecture''', the '''Ulam conjecture''' (after Stanislaw Ulam ), the '''Syracuse problem''', or the '''hailstone sequence''' or '''hailstone numbers'''. It asks whether a certain kind of number sequence always ends in the same way, regardless of the starting number.
said about the Collatz conjecture: "Mathematics is not yet ready for such problems." He offered $500 for its solution.
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 function f as follows:
Now, form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next.
In notation:
The Collatz conjecture is: ''This process will eventually reach the number 1, regardless of which positive integer is chosen initially.'' Or, more rigorously:

83. [ ]+[[-]+ [- ++ +[ ]++[- + ]] ] ++++++
The collatz problem or 3n+1 problem is as follows. Take a natural number n. If it s even, halve it; if odd, triple it and add one.
http://www.hevanet.com/cristofd/brainfuck/collatz.b
<, ] [The Collatz problem or 3n+1 problem is as follows. Take a natural number n. If it's even, halve it; if odd, triple it and add one. Repeat the process with the resulting number, and continue indefinitely. If n is 0, the resulting sequence is 0, 0, 0, 0... It is conjectured but not proven that for any positive integer n, the resulting sequence will end in 1, 4, 2, 1... See also http://www.research.att.com/projects/OEIS?Anum=A006577 This program takes a series of decimal numbers, followed by linefeeds (10). The entire series is terminated by an EOF (0 or "no change"). For each number input, the program outputs, in decimal, the number of steps from that number to zero or one, when following the rule above. It's quite fast; on a Sun machine, it took three seconds for a random 640-digit number. One more note. This program was originally written for Tristan Parker's Brainfuck Texas Holdem contest, and won by default (it was the only entry); the version I submitted before the contest deadline is at http://www.hevanet.com/cristofd/brainfuck/oldcollatz.b Daniel B Cristofani (cristofdathevanetdotcom) http://www.hevanet.com/cristofd/brainfuck/]

84. Collatz Problem
Translate this page The summary for this Japanese page contains characters that cannot be correctly displayed in this language/character set.
http://www.tekipaki.jp/~rootzx/html/Collatz Problem.html
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85. Collatz 3n+1 Problem Structure
Observations posted by Ken Conrow to stimulate further research.
http://www-personal.ksu.edu/~kconrow/
Ken Conrow Home Page
Collatz 3n+1 Problem Structure
As of early April 2005, I've again exhausted what I have to say about the Collatz conjecture, including pages about paradoxes encountered in this work, some discussion of Feinstein's proof of the unprovability of the Collatz conjecture, state transition diagrams to clarify the restrictions on the Collatz transitions, a revised slide show, and numerous additions and corrections. I am ready to claim a proof at this time based on a graph theoretic argument: a binary tree, but no other kind of graph, can be constructed from the nodes representing the odd positive integers. The nodes and their edges are based on the state transition diagram which describes the Collatz trajectories. Showing that the predecessor graph is a tree constitutes a proof of the Collatz conjecture. Mathematicians who refer to the problem as the problem were never brainwashed by FORTRAN (as I was) into the belief that n , not x , stands for an integer. I hope someone who can formalize mathematical proofs will see the potential here and take the appropriate set of ideas and sketch or complete a formal proof of the conjecture using them.

86. On The 3x + 1 Problem
Project working on the 3x+1 (or collatz) sequences and their properties.
http://personal.computrain.nl/eric/wondrous/

87. Number Of Iterations In Collatz's Problem - Maple Application Center - Maplesoft
Number of iterations in collatz s problem This sequence, attributed to Lothar collatz, has bee given various names, including Ulam s conjecture,
http://www.maplesoft.com/applications/app_center_view.aspx?AID=224

88. Collatz-Problem - Wikipedia
problems Wenn man das collatz-problem von den natürlichen Zahlen auf die ganzen
http://de.wikipedia.org/wiki/Collatz-Problem
Collatz-Problem
aus Wikipedia, der freien Enzyklop¤die
Inhaltsverzeichnis
Problembeschreibung L¶sungsans¤tze
Collatz-Problem in den ganzen Zahlen Verallgemeinerungen des Problems ...
Bearbeiten
Problembeschreibung
Das Collatz-Problem geh¶rt zu den ungel¶sten Problemen der Mathematik . Es wird gelegentlich auch 3n+1-Problem Syracuse-Vermutung Kakutanis Problem Hasse-Algorithmus Thwaites Vermutung oder Ulams Problem (nach Stanislaw Marcin Ulam ) genannt. Das Problem lautet: Man beginne mit einer beliebigen nat¼rlichen Zahl a und bilde damit die rekursive Zahlenfolge
Die Folge endet, wenn sie den Wert 1 erreicht. Alternativ reicht es auch, dass die Folge einen Wert der Form 2 n erreicht, da alle Zahlen der Form 2 n bei 1 enden. Vermutung : F¼r jede nat¼rliche Zahl a erreicht die Folge nach endlich vielen Schritten den Wert 1. Diese Vermutung konnte bisher weder bewiesen noch widerlegt werden. Beispiel: Sei a = 5. Dann erh¤lt man die Folge 5,16,8,4,2,1. F¼r a = 7 lautet die Folge 7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1.
Da auf jede ungerade Zahl n = 2k+1 eine gerade Zahl folgt ( 3n+1 = 3*(2k+1) +1 = 6k + 4 ist gerade) betrachtet man oft auch das ¤quivalente Problem Das Problem wurde von Lothar Collatz ver¶ffentlicht. Seitdem haben sich viele Mathematiker mit dem Problem besch¤ftigt. Mehrfach wurden Preise f¼r eine L¶sung ausgelobt.

89. P Andaloro's Abstract
collatz s problem The Venus Fly Trap of Mathematics. By Dr. Paul Andaloro Assistant Professor Department of Mathematics
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

90. InfoSatellite.com - The Collatz Conjecture
Erdös said about the collatz conjecture Mathematics is not yet ready for such problems . So asks Lagarias is the 3x+1 problem intractably hard?
http://www.infosatellite.com/news/2003/01/p070103collatz.html
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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 The site pass.maths.org.uk says that "These are sometimes called 'Hailstone sequences' because they go up and down just like a hailstone in a cloud before crashing to Earth - the endless cycle 4, 2, 1, 4, 2, 1. It seems from experiment that such a sequence will always eventually end in this repeating cycle 4, 2, 1, 4, 2, 1,... and so on, but some values for N generate many values before the repeating cycle begins. For example, try starting with n = 27. See if you can find starting values that generate even longer sequences". And it puts a slot so you can enter your chosen number (27, in this case) to be investigated, and the resulting chain of integers is given in a neat presentation:

91. Math 696 The 3x+1 Problem
called the 3x+1 problemabout the iterates of the collatz function. Is it the case that for every positive integer n, iterating the collatz function
http://www.math.tamu.edu/~harold.boas/courses/math696/Maple-3x 1.html

92. 15-121: Assignment 4
The socalled collatz function is a deceptively simple function defined on The problem remains open, despite considerable efforts by a large number of
http://ed.oc7.org/classes/15-121/lab4/lab4.htm
15-121: Java Prg
Assignment 4: The 3x+1 Problem
Due: Monday 12:01 AM, September 16, 2001
Description
The so-called Collatz function is a deceptively simple function defined on the positive integers:
coll( 1 ) = 1
coll( x ) = x/2 if x is even
coll( x ) = (3x+1)/2 if x is odd
It is straightforward to write a Java class that implements coll . A single application of coll is not particularly interesting, but when one applies the function repeatedly something strange happens: after a while one always reaches 1 (the fixed point). For example, starting at 18 we get Likewise, starting at 100 we get Of course, these are just two examples, but one can try out many more starting values, always with the same result: ultimately one reaches 1. This leads to the following conjecture. Conjecture: Starting at any positive integer, repeated application of the Collatz function will ultimately produce 1. The problem remains open, despite considerable efforts by a large number of people, and endless amounts of CPU time dedicated to it. The question is annoyingly simple to state, which makes it so particularly attractive somewhat like Fermat's last theorem, though that was recently solved. Any implementation of the Collatz 3x+1 function in Java using ordinary ints is really too feeble for interesting experiments: we need to be able to cope with large numbers. As it turns out, numbers of the form x = 2^k - 1 are particularly interesting. Your code should cope nicely with values of k around 5000.

93. THE INTRO TO AI SHOW ;;; Summer Session I 1999 ;;; ;;; DEMO
collatz s problem used to show loop and then ;;; dotimes. Is this a total function on the natural numbers? (defun mystery (n) (loop (print n) (when (or
http://www.rpi.edu/~brings/locker/collatz.lisp

94. CodeGuru: C++ Math And Fun
This problem was posed by L. collatz in 1937. In ther words, it is a very simple problem. Take any number as an input, and divide it by 2 if the number is
http://www.codeguru.com/Cpp/Cpp/algorithms/math/article.php/c7979/
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95. Collatz
Other collatzlike problems exist. For example, consider either of the two collatz s problem can be reduced to every integer becomes a number of the
http://www.users.globalnet.co.uk/~perry/maths/collatz/collatz.htm
Collatz
[Home] [Other Maths]
Collatz
Collatz's Conjecture says that every positive integer finally ends up at 1 when the following function is recursively applied: The results from this function are erratic. You can try some values out [here] . Pressing auto automatically increases the value by 2 and displays the results. This program displays the numbers generated in bases 2 through to 7, and base 10. Statistics There are loads of statistics to collect about the numbers generated by the Collatz function, for example, the length of a chain, the number of odd numbers in a chain, the number of primes generated in a chain, and so on. Here are a few: Length of chain Odd's generated Number of primes
Reduced Division An interesting Collatz technique goes like this: Count the number of iterations needed to get to one, but when counting count a run of divisions as only 1. e.g. to get from 16>1 is only counted as 1 step. As the solution to Collatz is tied up with the fact that the pattern of bits must tend to a single bit. As the division part just removes end zeros, and doesn't change the bit pattern, then we might aswell do all these steps in one go. See this [here] Other forms of the Collatz function Other Collatz-like problems exist. For example, consider either of the two mappings:

96. The Code Project - CPP Math And Fun - C++ / MFC
This problem is posed by L. collatz in 1937. In words, it is very simple problem. Take any number as an input, and divide it by 2 if no is even,
http://www.codeproject.com/cpp/CPPMathFun.asp

97. The Collatz (3x + 1) Problem -- From Mathematica Information Center
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
http://library.wolfram.com/infocenter/Demos/153/
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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
URL
http://mathworld.wolfram.com/CollatzProblem.html Downloads Collatz.m (4 KB) - Mathematica Package Files specific to Mathematica 2.2 version: Collatz.m (4 KB) - Mathematica Package Sign up for our newsletter:

98. The Collatz Sequence
An algorithm given by Lothar collatz produces sequences of integers, In this problem we want to determine the length of the sequence that includes all
http://acm.uva.es/p/v6/694.html

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).
Input
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

99. Problem 1 The Collatz Sequence
problem 1 The collatz Sequence. An algorithm given by Lothar collatz produces 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.

100. 3x+1 Conjecture Verification Results
The 3x+1 conjecture 1, 2, problem E16 asserts that starting from any 4, JC Lagarias and A. Weiss, The 3x+1 problem two stochastic models,
http://www.ieeta.pt/~tos/3x 1.html

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