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38
On the Collatz Problem
"... An attempt to come closer to a resolution of the Collatz conjecture is presented. The central idea is the formation of a tree consisting of positive odd numbers with number 1 as root. Functions for generating the tree from the root are presented and paths from nodes to the root are given by Collatz ..."
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by Collatz sequences. The Collatz problem is thus reduced to showing that all positive odd numbers are present in the tree.
The Undecidability of the Generalized Collatz Problem
, 2006
"... The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple iterative process halts on all inputs. We build on earlier work by J. H. Conway, and show that a natural generalization of the Collatz problem is recursively undecidable. 1 ..."
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The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple iterative process halts on all inputs. We build on earlier work by J. H. Conway, and show that a natural generalization of the Collatz problem is recursively undecidable. 1
The dual Collatz problem
, 2007
"... The generalized Collatz problem is defined by a sequence of natural numbers, generated conditionally by xn+1 = 1 2 xn if xn is even and by xn+1 = ..."
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The generalized Collatz problem is defined by a sequence of natural numbers, generated conditionally by xn+1 = 1 2 xn if xn is even and by xn+1 =
Article 08.4.7 The Collatz Problem and Analogues
"... In this paper, we study a polynomial analogue of the Collatz problem. Additionally, we show an additive property of the Collatz graph. 1 ..."
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In this paper, we study a polynomial analogue of the Collatz problem. Additionally, we show an additive property of the Collatz graph. 1
The 3n+1 Collatz Problem and Functional Equations
, 1995
"... This paper reports on some functional equations, which arise in connection with the famous 3n + 1 Collatz problem. In particular, it reports on two analytic versions of the corresponding Collatz conjecture, which are contained in [1]. ..."
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This paper reports on some functional equations, which arise in connection with the famous 3n + 1 Collatz problem. In particular, it reports on two analytic versions of the corresponding Collatz conjecture, which are contained in [1].
Nontrivial loops do not exist in the Collatz problem
, 907
"... In the Collatz problem, there are 3 possibilities. Starting from any positive number we either reach the trivial loop (1,4,2), end up in a nontrivial loop, or go till infinity. We will show here that there are no nontrivial loops other than the trivial (1,4,2) for positive integers. Also we shall ..."
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In the Collatz problem, there are 3 possibilities. Starting from any positive number we either reach the trivial loop (1,4,2), end up in a nontrivial loop, or go till infinity. We will show here that there are no nontrivial loops other than the trivial (1,4,2) for positive integers. Also we shall
A fractal set associated with the Collatz problem
"... In 1930’s, Lothar Collatz had great interest in representation of integer functions by directed graphs. He proposed the following, known as the HasseCollatzSyracuse problem: Problem 1.1. For a natural number n ∈ N, let us consider the function ..."
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In 1930’s, Lothar Collatz had great interest in representation of integer functions by directed graphs. He proposed the following, known as the HasseCollatzSyracuse problem: Problem 1.1. For a natural number n ∈ N, let us consider the function
c ○ SpringerVerlag 2000 Some results on the Collatz problem
, 1999
"... Abstract. The paper refersto the Collatz’s conjecture. In the first part, we present some equivalent forms of this conjecture and a slight generalization of a former result from [1]. Then, we present the notion of “chain subtrees” in Collatz’s tree followed by a characterization theorem and some sub ..."
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Abstract. The paper refersto the Collatz’s conjecture. In the first part, we present some equivalent forms of this conjecture and a slight generalization of a former result from [1]. Then, we present the notion of “chain subtrees” in Collatz’s tree followed by a characterization theorem and some
Cyclic Structure of Dynamical Systems Associated with 3x+d Extensions of Collatz Problem.
, 2000
"... . We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems D d generated by iterations of the functions T d acting, for all d # 1 relatively prime to 6, on positive integers : T d : N # N; T d (n) = n 2 ..."
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. We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems D d generated by iterations of the functions T d acting, for all d # 1 relatively prime to 6, on positive integers : T d : N # N; T d (n) = n 2 , if n is even; 3n+d 2 , if n is odd. In the case d = 1, the properties of the system D = D 1 are the subject of the wellknown 3x + 1 conjecture. For every one of 6667 systems D d , 1 # d # 19999, we calculate its (complete, as we argue) list of primitive cycles. We unite in a single conceptual framework of primitive memberships, and we experimentally confirm three primitive cycles conjectures of Je# Lagarias. An indeep analysis of the diophantine formulae for primitive cycles, together with new rich experimental data, suggest several new conjectures, theoretically studied and experimentally confirmed in the present paper. As a part of this program, we prove a new upper bound t...
Results 1  10
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38