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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

lists with details

The 3n+1 Collatz Problem
The 3n+1 collatz problem. This problem has a mysterious history andit is not easy to explain the attraction of mathematicians for it.
http://www.ime.usp.br/~mane/collatz.html

Extractions: This problem has a mysterious history and it is not easy to explain the attraction of mathematicians for it. Consider f(n)=n/2 in n is even and f(n)=(3n+1)/2 if n is odd. It is true that, for all n, the orbit of n by f ends in the cycle (1-2)? The answer for this question is unknown and a lot of problems have borned in the research of this answer (e.g. it is true that the conjecture holds for a set A of positive density? (answer: unknown). Or, if the orbit of n by f is bounded it ends in the cycle (1-2)? (answer: unknown)). We study several aspects of the dynamical system defined by function f and several generalizations of this function. Manuel Valentim de Pera Garcia - (e-mail mane@ime.usp.br Fabio Armando Tal - (email fabiotal@ime.usp.br A note on the generalized 3n+1 problem - Manuel V. P. Garcia and Fabio A. Tal - Acta Arithmetica XC 3 (1999) pp. 245-250 mane@ime.usp.br Up to research fields in applied math
Mathenomicon.net : Reference : Collatz Problem
collatz problem. noun. The collatz problem is as follows Take any x Î %Nset(); % ; If x is odd, set x to be 3x + 1. Otherwise, set x to be x/2;
http://www.cenius.net/refer/display.php?ArticleID=collatzproblem
The Collatz Problem
The collatz problem. This is one of the unsolved problems in Mathematics.You start with a positive integer. If it is even, you halve
http://www.maths.ex.ac.uk/pumc/collatz.html

Extractions: This is one of the unsolved problems in Mathematics. You start with a positive integer. If it is even, you halve it; if it is odd you multiply by 3 and add 1. You repeat the rule on the result and keep on going, repeating the rule on that result and so on. The conjecture is that you will finish up at 1. E.g. However, nobody has managed to prove this yet. Try playing about with this simple idea and check your sequences with this form Type your starting value for the sequence
PlanetMath: Collatz Problem
collatz problem, (Conjecture). We define the function so that. is an arbitraryseed value. collatz problem is owned by akrowne. (view preamble)
http://planetmath.org/encyclopedia/142Problem.html

Extractions: Collatz problem (Conjecture) We define the function so that is an arbitrary seed value. It is conjectured that the sequence will always end in , which repeats infinitely. This has been verified by computer up to very large values of , but is unproven in general. It is also not known whether this problem is decideable. This is sometimes called the ``hailstone sequence'' because, like a hailstone in a cloud, the values oscillate up and down.
Collatz Problem
An Image from the collatz problem The intensity of a point denotes the time takento terminate. collatz problem - From Eric Weissten's World of Mathematics.
http://numbersorg.com/NumberTheory/CollatzProblem/

Extractions: Search The Net Top Science Math Number Theory : Collatz Problem The Structure of the 3x + 1 Function - A paper by Peter Schorer describing a new approach to the 3x + 1 Problem and indicating a possible solution. 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. An Image from the Collatz Problem - The intensity of a point denotes the time taken to terminate. 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.
Collatz Problem Information Sites
Reviewed collatz problem sites, by people who know collatz problem and work withcollatz problem. collatz problem From Eric Weissten's World of Mathematics.
http://numbersorg.com/NumberTheory/OpenProblems/CollatzProblem/

Extractions: 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 - Katholische Universit¤t Eichst¤tt, Germany; 56 August 1999. On-line proceedings and group photo.
The Proof Of The Collatz Problem
The proof of the collatz problem. The proof by collatz function. a n /2, if an =0 (mod 2). The third step is practiced the collatz problem. Then By My Formula.
http://www.angelfire.com/journal2/collatz/Proof.htm

Extractions: The proof of the Collatz Problem The proof : by collatz function a n if a n (mod 2) a n n if a n (mod 2) a n+1 2*s-1 +s)*a n +s , a n =s (mod 2) The second step is generated the function a n+1 +s )*a n +s , a n =s (mod 2 a n+1 +s )*a n +s +s )+s a n =s (mod 2) We can find a general formula for this expression Then By My Formula and we omit it because a n dosent equal zero And 2*s(j-i)-1 doesnt equal zero. and The forth step we suppose that at ,a n+1 =1 if it holds a n+1 at any number less than The fifth step is analysed a n =1 and =1 it will be omit because s j- i =0 either or 1 only a n and ,s j-i =0 then a n =2 and it is right But is 1= And by Terras (1976, 1979) function t n if t n =0 (mod 2) t n+1 n if t n =1 (mod 2) We find that for all 1 step by 3t n +1 there is two steps by divide on 2 then (1/3)j= And then and Collatz conjecture be right.
Bookmarks 3xp1 Problem Syracuse Collatz
superLagarias.html The Collatz (3x + 1) Problem Ilan Vardi http//www.mathsource.com/Content/Applications/Mathematics/0200305 collatz problem http//www
http://perso.wanadoo.fr/patrick.davalan/Liens/liens_syracuse.html
Collatz Problem
collatz problem. A Conjecture. Let be an Integer. Then the collatz problemasks if iterating, (1). always returns to 1 for Positive .
http://lib4web.lib.msu.edu/crcmath/math/math/c/c433.htm

Extractions: A problem posed by L. Collatz in 1937, also called the x +1 Mapping Hasse's Algorithm Kakutani's Problem ... Thwaites Conjecture , and Ulam's Problem (Lagarias 1985). Thwaites (1996) has offered a 1000 reward for resolving the Conjecture . Let be an Integer . Then the Collatz problem asks if iterating always returns to 1 for Positive . This question has been tested and found to be true for all numbers (Leavens and Vermeulen 1992), and more recently, (Vardi 1991, p. 129). The members of the Sequence produced by the Collatz are sometimes known as Hailstone Numbers Negative numbers are included, there are four known cycles (excluding the trivial cycle): (4, 2, 1), ( ), and ( ). The number of tripling steps needed to reach 1 for , 2, ... are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane's
AMCA: The Relation Between Collatz Conjecture And Primes Presented By Murad A. A
the wellknown conjecture ,Collatz conjecture One of many conjectures still withoutproof a problem posed by L. Collatz in 1937 The collatz problem asks if
http://at.yorku.ca/cgi-bin/amca/caft-16
Science/Math/Number_Theory/Collatz_Problem
The Collatz, Syracuse or 3x+1 problem / Science /Math / Number_Theory / collatz problem. The collatz problem is a special case.Search Category
http://www.science-and-research.com/Science/Math/Number_Theory/Collatz_Problem/
Www-irma.u-strasbg.fr/irma/seminaires/autres/ACP-2001.shtml
Similar pages IRMA Strasbourg Publication 2000 Belaga,Maurice Mignotte, (fichier Postscript compressé, 792 Ko) Cyclic Structureof Dynamical Systems Associated with $3x+d$ Extensions of collatz problem..
http://www-irma.u-strasbg.fr/irma/seminaires/autres/ACP-2001.shtml
Introduction To Artificial Intelligence Selmer Bringsjord
Hyperproof file.); A picture of the collatz problem; More on the CollatzProblem and Related Problems; Is English Compositional? (picture
http://www.rpi.edu/~brings/intai.html

Extractions: The Intro to AI Show Meet the Cast Video Highlights Handouts ... Supporting Links Problems/Puzzles for Smart Agents 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).
Www.math.niu.edu/~rusin/known-math/99/collatz_undec
seem to recall that Conway may have established this fact.) Conway defined a familyof problems, a natural generalization of the collatz problem, and showed
http://www.math.niu.edu/~rusin/known-math/99/collatz_undec
Www.math.niu.edu/~rusin/known-math/98/collatz_hide
From Bill Daly billdaly@interport.net Newsgroups sci.math.research,sci.mathSubject Reformulation of collatz problem Date Wed, 4 Mar 1998 121004
http://www.math.niu.edu/~rusin/known-math/98/collatz_hide
Sci.math FAQ: Unsolved Problems
collatz problem * Goldbach's conjecture * Twin primes conjecture _Names of large numbers
http://isc.faqs.org/faqs/sci-math-faq/unsolvedproblems/

Extractions: Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76LD.Fnt@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math hv@cix.compulink.co.uk (Hugo van der Sanden): To the best of my knowledge, the House of Commons decided to adopt the US definition of billion quite a while ago - around 1970? - since which it has been official government policy. dik@cwi.nl (Dik T. Winter): The interesting thing about all this is that originally the French used billion to indicate 10^9, while much of the remainder of Europe used billion to indicate 10^12. I think the Americans have their usage from the French. And the French switched to common European usage in 1948. gonzo@ing.puc.cl alopez-o@barrow.uwaterloo.ca By Archive-name By Author ... Help
F. Conjectures (Math 413, Number Theory)
A collection of easily stated conjectures which are still open. Each conjecture is stated along with Category Science Math Number Theory Open Problems Conj All integers greater than 1 have occur as multiplicities. (Sierpinski) RefsUnsolved Problems in Number Theory, Guy, 1994, Sect B.37. The collatz problem.
http://www.math.umbc.edu/~campbell/Math413Fall98/Conjectures.html

Extractions: 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 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. Conj: The only zeros of Z( s ) are at s 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. 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 n , is defined by M n n Thm: M n is prime implies that n n is perfect. (Euclid)
Numcom08
NUMBERS AND COMPUTERS (8) by Albert N. Debono THE collatz problem. Takeany positive number. If it is odd, multiply it by three and add one.
http://www.eng.um.edu.mt/~andebo/numbers/numcom08.htm

Extractions: 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.
Artslist
Numbers and Amicable Pairs 4. The Prime Numbers 5. Aliquot Sequences 6. Abundantand Deficient Numbers 7. Kaprekar Numbers 8. The collatz problem 9. Mersenne
http://www.eng.um.edu.mt/~andebo/numbers/artslist.htm
Math Forum - Ask Dr. Math
What's the difference between problems like Squaring the Circle andGoldbach's Conjecture or the collatz problem? Drexel dragon
http://mathforum.org/library/drmath/view/52540.html

Extractions: Associated Topics Dr. Math Home Search Dr. Math Date: 02/19/98 at 18:16:57 From: D-Jitsu Subject: 11th Grade Research Paper Dear Dr. Math, I need help getting started on a research paper, as well as on doing the project. The topic of my report is "The Impossible and Unsolved in Mathematics (Goldbach's Conjecture, Squaring the Circle, etc.). I was just wondering if I could get some references, tips, or ideas on how to get started. This is for a pre-calculus class. http://ug.cs.dal.ca/~campbell/colllink.html http://mathforum.org/dr.math/ Associated Topics

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