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Topic: Collatz conjecture

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	Collatz conjecture - Wikipedia, the free encyclopedia
		The Collatz conjecture is an unsolved conjecture in mathematics.
		The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), the Syracuse problem, as the hailstone sequence or hailstone numbers, or as Wondrous numbers as per Gödel, Escher, Bach.
		Collatz Observations discusses observations about the Collatz Conjecture as it pertains to a bijection over the natural numbers.
	en.wikipedia.org /wiki/Collatz_conjecture (1742 words)

	Conjecture - Wikipedia, the free encyclopedia
		In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove.
		Once a conjecture has been proven, it becomes known as a theorem, and it joins the realm of known mathematical facts.
		Although many of the most famous conjectures have been tested across an astounding range of numbers, this is no guarantee against a single counterexample, which would immediately disprove the conjecture.
	en.wikipedia.org /wiki/Conjecture (680 words)

	PlanetMath: Collatz problem (Site not responding. Last check: 2007-11-07)
		This is version 16 of Collatz problem, born on 2001-08-19, modified 2003-11-05.
		Basically, all the orbits of the Collatz function form a basis for a topology on the positive integers induced by this function.
		A corollary to a theorem states that the Collatz Conjecture is true if and only if the topology induced by the Collatz Function on the positive integers is connected.
	planetmath.org /encyclopedia/CollatzProblem.html (192 words)

	Conjecture: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-07)
		In mathematics, goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics....
		The collatz conjecture is an unresolved conjecture in mathematics....
		In mathematics, the poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds....
	www.absoluteastronomy.com /enc3/conjecture (1972 words)

	Collatz (Site not responding. Last check: 2007-11-07)
		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.
		Collatz's problem can be reduced to every integer becomes a number of the form 11..11 in base 4.
		Assume this n is a candidate for the falsehood of the Collatz conjecture.
	www.users.globalnet.co.uk /~perry/maths/collatz/collatz.htm (742 words)

	The 3x+1 problem and its generalizations (Site not responding. Last check: 2007-11-07)
		The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of the iterates of the function which takes odd integers n to 3n+1 and even integers n to n/2.
		The 3x+1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1.
		The 3x+1 Conjecture is simple to state and apparently intractably hard to solve.
	www.cecm.sfu.ca /organics/papers/lagarias (170 words)

	Collatz conjecture (Site not responding. Last check: 2007-11-07)
		All the elements of the same Collatz's sequence are said to be C-equivalent.
		A conjecture is made that every natural number n is C-equivalent to a natural number m < n, i.e.
		Even though this conjecture has been tested for many numbers by calculation programs like the applet at the beginning of this page, it is not generally demonstrated for every natural number.
	www.robertobigoni.it /Matematica/collatz/collatzeng.html (624 words)

	Collatz conjecture - QuickSeek Encyclopedia (Site not responding. Last check: 2007-11-07)
		The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), the Syracuse problem, as the hailstone sequence or hailstone numbers, or as Wondrous numbers as per Gödel, Escher, Bach.
		The Collatz map can be looked as a restriction to the integers of the smooth real and complex map
		3m + 1 is guaranteed to be in the Collatz sequence of 4m + 1.
	collatzconjecture.quickseek.com (1823 words)

	[No title]
		The conjecture involves a recursive iteration on the positive integers and proposes that the trajectory of these iterations is such that convergence to 1 will result from any starting value.
		By regarding the Collatz iteration as a finite state machine, a directed graph called a state transition diagram is readily constructed.
		Note: the predecessor direction is from a node to its Collatz predecessor, eqn 2, opposite to that of the Collatz trajectory, eqn 1.
	www-personal.ksu.edu /~kconrow/concise.html (1859 words)

	3x+1 conjecture verification results
		The 3x+1 conjecture [1], [2, problem E16] asserts that starting from any positive integer n the repeated iteration of T(x) eventually produces the integer 1, after which the iterates will alternate between the integers 1 and 2.
		In order to test the 3x+1 conjecture, in 1996 we wrote a computer program (in the programming language C), which computed the trajectories of all initial values of n smaller that a given limit and having a stopping time known to be larger than 40 [3].
		Since no counter-example was found, the 3x+1 conjecture is probably true for all positive integers not larger than this verification limit.
	www.ieeta.pt /~tos/3x+1.html (742 words)

	InfoSatellite.com - The Collatz Conjecture
		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, adds mapping problem, Hasse's algorithm, Kakutani's problem, Syracuse algorithm, Syracuse problem, and Thwaites conjecture.
		Paulos explains that The Collatz conjecture is that no matter what number you choose, you'll always end up with 4, 2, 1, 4, 2, 1,....
		One of the greatest numbers subjected to the Collatz conjecture that had its algorithm subjected to computerized calculation was described by its author in Oliveira e Silva, T. "Maximum Excursion and Stopping Time Record-Holders for the 3x+1 Problem: Computational Results".
	www.infosatellite.com /news/2003/01/p070103collatz.html (861 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Daniel Bernstein: A non-iterative 2-adic statement of the 3x+1 conjecture.
		Franco/Carl Pomerance: On a conjecture of Crandall concerning the QX+1 problem.
		3283 Benedict Seifert: On the arithmetic of cycles for the Collatz-Hasse ('Syracuse') conjectures.
	felix.unife.it /Root/d-Mathematics/d-Number-theory/b-3x+1 (460 words)

	Re: Primes and the Collatz conjecture!
		Where each node off this path is >>only one and that is where all of these special odd primes and 0 (mod) >>3's and other odd composites with prime factors reside thus creating >>only 3 odd integers, the (seed), prime 5, and 1 in their entire path >>starting with (seed) 3.
		You also have to >>consider the 10 factor of the Collatz* (seed) primes when comparing >>densities of the two.
		Yes, there is many more even exponents creating the >>Collatz (seed) prime then the Mersenne prime exponents, so that will >>explain the density factor between the two.
	www.usenet.com /newsgroups/sci.math/msg00584.html (590 words)

	Math Conjectures (Site not responding. Last check: 2007-11-07)
		A study to be released today looking at long-term trends in test scores and academic success argues that widespread reports of U.S. boys being in crisis are greatly overstated and that young males in school are in many ways doing better than ever.
		Each conjecture is stated along with a collection of references.
		One of the things that turned me on to math were some simple sounding but unsolved problems that were easy for a high school student to understand.
	finitegeometry.info /info/Math-Conjectures (255 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		To solve the Collatz-like conjectures, we have only to identify the x's to which such a sequence can converge _when starting with a natural number_.
		To: rusin@math.niu.edu Subject: Re: Collatz "proof" (was Re: Collatz ("3n+1") Problem: discoveries and questions) well, the extension i had in mind was i thought the obvious entire complex-analytic extension: form a linear combination of "3z+1" and "z/2" with appropriate "trigonometric" coefficients, if you know what i mean.
		To: rusin@math.niu.edu Subject: Re: Collatz "proof" (was Re: Collatz ("3n+1") Problem: discoveries and questions) Well, to start with, you have a lot more fixed points!, roughly near each large solution of tan(pi*z/2)=1/2.
	www.math.niu.edu /~rusin/papers/known-math/98/collatz_mine (1593 words)

	Collatz 3n+1 Problem Structure
		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.
		Using the symbols {s,b,e} to denote fragments of the paths in the binary predecessor tree and encoding those as {'10','11','0'} achieves a shorter bit string representation of a path than the parity bit string.
	www-personal.ksu.edu /~kconrow (1785 words)

	sciforums.com - Collatz Conjecture
		The conjecture states that for any number in Z+ the sequence will end in the cycle 4, 2, 1.
		It is a very interesting problem, so much so that Paul Erdos thought it was beyond the mathematics of his time.
		I worked on the conjecture for about three days last year.
	www.sciforums.com /printthread.php?t=48836 (750 words)

	David - Collatz conjecture
		In this paper, we show that any proof of the Collatz 3n+1 Conjecture must have an infinite number of lines; therefore, no formal proof is possible.
		Finally, here's the first link I looked at when trying to remember the name of the "Collatz Conjecture": Article about 3 unsolved math problems.
		A copy of an article from elsewhere, it includes mention of a couple of other fun problems as well, and even promises a $1 million prize for a solution (to be delivered on the second Tuesday of the week in which the solution is announced).
	www.crazylife.org /~util/133489.html (183 words)

	The Collatz Conjecture
		The following is a graph of the solution space to the Collatz problem.
		Perhaps this isn’t as big a question as was originally thought.
		If no such proof exists, and we prove the Collatz conjecture by other means, this will also be a proof that all natural numbers are represented by (2k + 1) * 2^p.
	www.rankyouragent.com /primes/collatz.htm (1071 words)

	List of conjectures - ExampleProblems.com
		Erdős conjecture, which lists conjectures of Paul Erdős and his collaborators
		Blattner's conjecture (now often known as the Blattner formula)
		Epsilon conjecture (an intermediate on the way to Fermat's last theorem)
	www.exampleproblems.com /wiki/index.php/List_of_conjectures (95 words)

	Math Forum Discussions
		Q:The Mersenne Primes and the Collatz conjecture (3x+1)/2.
		Re: Q:The Mersenne Primes and the Collatz conjecture (3x+1)/2.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?threadID=1311039&messageID=4138455 (194 words)

	How to Prove The Collatz Conjecture - book profile from the WritersNet books and publications directory
		I proved the Collatz Conjecture, an unsolved problem in mathematics until now.
		I included a computer program that proves the Collatz Conjecture beyond a doubt.
		The Collatz Conjecture claims that any positive integer will reach 1 if it is divided by 2 (x/2) when even and multiplied by 3 with 1 added if odd (3x+1).
	www.writers.net /writers/books/21179 (99 words)

	Re: Thoughts on the Collatz conjecture
		This "idea" I'm thinking of needs to be kicked around a bit so, I hope Collatz people will give it a kick or two before it gets posted to the world on Google.
		My "partner" is into "higher math studies" so he would like to know how it goes but doesn't have the time at this time to work with me on it.
		Re: Thoughts on the Collatz conjecture, Ernst Berg
	www.usenet.com /newsgroups/sci.math/msg01386.html (306 words)

	Citations: Acta Arith - Cohen, of, finite (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Of course (C) is equivalent to the conjunction of (A) and (B) A) 1,2) is the only Collatz cycle in N.) Every Collatz sequence in N is bounded.
		Of course (C) is equivalent to the conjunction of (A) and (B) A) 1,2) is the only Collatz cycle in IIN.
		, 9] 6] 3] and [10] Partial results are: D) The Collatz conjecture holds true for all initial values x 0 6:3 10 13 (see [7] E) The length of a Collatz cycle in IIN which does not contain 1 is at least 17087 915 (see [3] 2 L. UHLER The aim of this article....
	citeseer.ist.psu.edu /context/135874/0 (505 words)

	Atlas: The relation between collatz conjecture and primes by Murad A. AlDamen (Site not responding. Last check: 2007-11-07)
		Atlas: The relation between collatz conjecture and primes by Murad A. AlDamen
		One of many conjectures still without proof a problem posed by L. Collatz in 1937
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caft-16.
	atlas-conferences.com /c/a/f/t/16.htm (142 words)

	My Project (Site not responding. Last check: 2007-11-07)
		So in the long run this searches at n(log n)^2, which should be quite fast, an a new way to look at the project.
		It is a very simple program made to progress the numbers in Collatz, however, it seems that people out there with more optimized algorithms and faster comptuers have gotten to numbers of around 2^300, way more than what I can do.
		Part of the reason I decided to drop this project.
	www.tjhsst.edu /~yzhang/techlab/proj2/index_collatz.html (382 words)

	Collatz Conjecture
		Hello people, I have some ideas about the collatz conjecture.
		Hi, I would like to here anything from people who have done some personal study on the Collatz.
		I am also interested in hearing about any other unproven c...
	groups.msn.com /CollatzConjecture (102 words)

	How to Prove The Collatz Conjecture by Danny Fleming (Book) in
		How to Prove The Collatz Conjecture by Danny Fleming (Book) in
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	www.lulu.com /browse/book_view.php?fCID=34570 (66 words)

	Puzzle - Collatz Conjecture
		For any positive integer N, will this algorithm always reach 1 eventually?
		A reader sent this to me, and I did not know that it is actually an open conjecture — the Collatz Conjecture.
		If you did solve it, you can claim a $1,000 prize!
	www.olimu.com /Notes/Collatz.htm (77 words)

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