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  1. OPTIMIZATION PROBLEMS.Translated by P.Wadsack.*(Applied Mathematical Sciences,17) by L./Wetterling,W. Collatz, 1975
  2. Optimization Problems (Applied Mathematical Sciences Ser. ; Vol. 17)) by Lothar Collatz, 1975-07
  3. 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
  4. Numerical Treatment of Eigenvalue Problems (International Series of Numerical Mathematics)
  5. Differential-Difference Equations: Applications and Numerical Problems (International Series of Numerical Mathematics) by L. Collatz, 1983-05
  6. Applied Mathematical Sciences, 17 Optimization Problems, Translated by P. Wadsack, by L., Collatz, 1975
  7. The Dynamical System Generated by the 3n+1 Function (Lecture Notes in Mathematics) by Günther J. Wirsching, 1998-03-20

41. Kaprekar Number
The 3n+1 Problem (collatz problem) top Take any natural number, from whichyou derive a sequence of numbers according to the following rules.
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)

42. Mathematical Problems - Problem Solving - Mathematical Competitions (Math Links
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;
Mathematical Problems - Problem Solving
Mathematical Competitions
not a complete list, only what I happened to see...

43. 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
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]

44. Atlas: The Relation Between Collatz Conjecture And Primes By Murad A. AlDamen
the wellknown conjecture ,Collatz conjecture One of many conjectures still withoutproof a problem posed by L. Collatz in 1937 The collatz problem asks if
Atlas Document # caft-16 Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics
December 21-24, 2000
University of Craiova
Craiova, Romania Conference Organizers
Minh Perez (American Research Press, Rehoboth, Box 141, NM 87301, USA) and Vasile Seleacu (University of Craiova, Department of Mathematics, Craiova, Romania)
View Abstracts
Conference Homepage The relation between collatz conjecture and primes
presented by
Murad A. AlDamen
University of Jerash Date received: October 4, 2000
The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc.

45. Internet Center For Mathematics Problems
collatz problem. Famous Problems in Mathematics The Four Colour Theorem;The Trisection of an Angle. Which are the 23 Hilbert Problems?
Internet Center for Mathematics Problems
On this page, we try to identify and list all sources of mathematics problems on the Internet and related information. Please advise us if you know of any sources that we are missing.
Table of Contents

46. 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
The Intro to AI Show
Your Host, Selmer Bringsjord and Selmer as....
Meet the Cast Video Highlights Handouts ... Supporting Links

47. From (Gerry Myerson) Subject Re Is Collatz
seem to recall that Conway may have established this fact.) Conway defined a familyof problems, a natural generalization of the collatz problem, and showed
From: (Gerry Myerson) Subject: Re: Is Collatz Conjecture (3N+1 Problem) Undecidable? Date: Wed, 30 Jun 1999 13:05:01 +1100 Newsgroups: sci.math Keywords: Conway's generalizations include some undecideable problems In article

48. From Robert Harrison Mongrels@streamlinerec.
odds. The 2^23^1 = 1 values for s and t is related to the know attractor of the collatz problem the 1- 4- 2- 1 cycle. Robert
From: "Robert Harrison" Newsgroups:,alt.tanaka-tomoyuki,sci.math Subject: Re: Fermat, Goldbach, open problems Date: Wed, 23 Dec 1998 19:59:27 -0000 steiner wrote in message ... >In article

49. 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 primenumbers = 3 have the same successor as they do in the Collatz sequence
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.

    50. The Collatz Problem, Also Known As The 3x+1 Problem
    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.
    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.
    <= Abs[n]

    51. F. Conjectures (Math 413, Number Theory)
    Conj All integers greater than 1 have occur as multiplicities. (Sierpinski). RefsUnsolved Problems in Number Theory, Guy, 1994, Sect B.37. The collatz problem.
    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
  • The Riemann Hypothesis
    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)

    52. E. Akin Why Is The 3x+1 Problem So Hard? Math. Magazine (ca.
    Eichsta t Conference 1999, 5p. 13469 Stefan Andrei/Manfred Kudlek/Radu Stefan NiculescuSome results on the collatz problem. Eichsta t Conference 1999, 15p. 1

    53. !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!
    Translate this page Antwort *grins*. Schauma mal, wer das zusammengebracht hat Jens löst das Rätselam 11.3.2003 um 1202 Uhr Es handelt sich dabei um das collatz problem.
    oder: Des Pudels Kern lautete:
    The 3n+1 problem
    Eingeschickt von niemandem
    Schauma mal, wer das zusammengebracht hat ...
    Jens < 3 * 2^53 = 2.7...*10^16 (Oliveira e Silva 1999) und Thwaites (1996) has offered a £1000 reward for resolving the conjecture. Das heisst, du willst die 1000 Pfund haben und derjenige, der die Arbeit macht bekommt nur ein Bier - so nicht ;-) Flo F Legende zur Bewertung
    Lili -
    Freitag, 26-Sep-2003 22:16:59 UTC

    54. [The Collatz Problem Or 3n+1 Problem Is As Follows. Take A Natural
    The collatz problem or 3n+1 problem is as follows. Take a natural numbern. If it s even, halve it; if odd, triple it and add one.]>,[ [ [ >>>[>>>>]+[[-]+ <, ] [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 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 Daniel B. Cristofani

    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);,,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 Mathsoft Constants ... Math Resources Unsolved Problems on Other Sites

    56. Index Of /muneer_ilyas/school/Maths Problems/Collatz Problem (£1000)_files
    Index of /muneer_ilyas/school/Maths problems Parent Directory 09Apr-2003 0815 - collatz problem (£10.. 21-Mar-2003 0414 36kcollatz problem (£10.. 21-Mar-2003 0346 - Help files/ 21-Mar-2003 0406 problems/Collatz Problem
    Index of /muneer_ilyas/school/Maths problems/Collatz Problem (£1000)_files
    Name Last modified Size Description ... Parent Directory 21-Mar-2003 05:04 - book.gif 21-Mar-2003 02:50 1k c2_253.gif 21-Mar-2003 02:55 1k c2_26.gif 21-Mar-2003 02:53 1k c2_39.gif 21-Mar-2003 02:54 1k c2_531.gif 21-Mar-2003 02:55 1k c2_532.gif 21-Mar-2003 02:56 1k c2_533.gif 21-Mar-2003 02:56 1k c2_534.gif 21-Mar-2003 02:57 1k c2_535.gif 21-Mar-2003 02:58 1k c2_536.gif 21-Mar-2003 02:58 1k c2_537.gif 21-Mar-2003 02:59 1k c2_538.gif 21-Mar-2003 03:00 1k c2_539.gif 21-Mar-2003 03:01 1k c2_540.gif 21-Mar-2003 03:03 1k c2_541.gif 21-Mar-2003 03:04 1k c2_542.gif 21-Mar-2003 03:06 1k c2_543.gif 21-Mar-2003 03:07 1k c2_544.gif 21-Mar-2003 03:08 1k c2_545.gif

    57. EDUCATION PLANET - 1228 Web Sites For Problem Solving
    12. The 3x+1 problem and its generalizations The 3x+1 problem, also known as thecollatz problem, the Syracuse problem, Kakutani s problem, Hasse s algorithm

    58. Math Trek: Dangerous Problems, Science News Online, June 29, 2002
    has baffled mathematicians for more than 70 years. Also known as thecollatz problem, it concerns a sequence of positive integers.
    Math Trek
    Dangerous Problems
    Food for Thought
    Slugging It Out with Caffeine
    Science Safari
    Remember Typewriters?
    70 Years Ago in
    Science News
    Week of June 29, 2002; Vol. 161, No. 26
    Dangerous Problems
    Ivars Peterson Some mathematical problems are easy to describe but turn out to be notoriously difficult to solve. Nonetheless, despite their reputed difficulty and repeated warnings from those who had failed to solve them in the past, these infamous problems continue to lure mathematicians into hours, days, and even years of futile labor. In a presentation this week on "mathematical problems between order and chaos," Jeffrey C. Lagarias of AT&T Labs–Research highlighted three such notorious unsolved problems. His was a cautionary tale, aimed at an audience that included 12 high-school students who had already shown their problem-solving proficiency by topping the 2002 U.S.A. Mathematical Olympiad (USAMO). Susceptible himself to the lure of these tantalizing conundrums, Lagarias admitted that he could have subtitled his talk "some problems I wish I could solve." In recent years, Lagarias has made important research contributions in a variety of mathematical fields, including work on the randomness of pi's digits, number patterns related to circles nested within circles, and the problem of distinguishing knots from unknots. Billiards in triangles In a game of mathematical billiards, a ball moves across a table of a given shape at a constant speed forever. There's no friction to slow the ball down. It simply travels in a straight line until it hits a cushion and rebounds according to the rule that the angle of reflection equals the angle of incidence.

    59. 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.
    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).

    60. Science: Mathematics: Number_Theory: Open_Problems - Open Site
    23 5 giving A = 1.6299. collatz problem. Also known as the 3n+1 problem,Syracuse problem, Thwaites problem. Define a function on
    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.

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