Results 11 to 20 of about 10,716 (231)
Theoretical and computational bounds for m-cycles of the 3n+1-problem [PDF]
Acta Arithmetica, 2005 Summary: An \(m\)-cycle of the \(3n+1\)-problem is defined as a periodic orbit with \(m\)~local minima. In this article we derive lower and upper bounds for the cycle length and the elements of (hypothetical) \(m\)-cycles. In particular, we prove that there do not exist nontrivial \(m\)-cycles for \(1\leq m\leq 68\).
Simons, J.L., de Weger, B.M.M.
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On heights in the Collatz 3n + 1 problem
Discrete Mathematics, 1985 The article describes the construction of pairs of consecutive integers which reduce to 1 for the first time in the same number of iterations of the function \(T(n)=n/2\), if \(n\) is even, and \(=(3n+1)/2\) if \(n\) is odd. The number of such pairs is shown to be infinite by means of free groupoids on the vectors of the parities of the values of the ...
Garner, Lynn E.
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Nepal Journal of Mathematical Sciences, 2020 The subject of this paper is the well-known 3n + 1 problem of elementary number theory. This problem concerns with the behaviour of the iteration of a function which takes odd integers n to 3n + 1, and even integers n to n/2. There is a famous Collatz conjecture associated to this problem which asserts that, starting from any positive integer n ...
Ajaya Singh, Bishnu Hari Subedi
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Computer-Based Validation of 3n+1 Hypothesis for Numbers 3n−1
Tehnički Vjesnik, 2019 The formulation of the 3n−1 problem is simple but no one has found the solution yet. This paper transforms the original problem into its equivalent so that it becomes more suitable for computer validation. A new algorithm is proposed and implemented. The
Tomovic, Savo +3 more
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A non-iterative 2-adic statement of the 3N+1 conjecture [PDF]
, 1994 Associated with the 3 N + 1 3N + 1 problem is a permutation Φ \Phi of the 2-adic integers.
Daniel J. Bernstein +2 more
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Collatz Conjecture (3N+1) Solution
Journal of Mathematics and Statistics StudiesCollatz Conjecture (3x+1) or in some literature as 3N+1 is a problem because it works in the way that if you take any positive number, if it is an odd number you multiply it by three (3) then add one (1).
AL-Shammary, Naif Mohammad
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On the problem of positive predecessor density in $3n+1$ dynamics
Discrete and Continuous Dynamical Systems, 2003 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
G. Wirsching
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Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3
, 2023 Collatz conjecture states that an integer $n$ reduces to $1$ when certain simple operations are applied to it. Mathematically, the Collatz function is written as $f^k(n) = \frac{3^kn + C}{2^{z}}$, where $z, k, C \geq 1$. Suppose the integer $n$ violates Collatz conjecture by reappearing as $2^in$, where $i \geq 1$, then the equation modifies to $n ...
Gaurav Goyal (15202048)
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Multiple positive solutions for a nonlinear 3n-th order three-point boundary-value problem
Electronic Journal of Differential Equations, 2010 In this article we establish the existence of at least three positive solutions for 3n-th order three-point boundary value problem by using five functional fixed point theorem.
K. L. Saraswathi Devi, Kapula R. Prasad
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Explanation of the Dynamics involved in the 3N+1 Problem A proof for The Collatz Conjecture
, 2021 The scope of the present work is to explain why it is true that all N have a distinct position in The Collatz Tree (The Collatz Graph)
Bruun, R.
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