Related papers: On the Boundedness of Collatz Sequences
In this paper we are shown the following facts: The probability of increased $ A_{k}=P(T^{k} (x_{0})>T^{k-1} (x_{0})) $, and the probability of decrease $B_{k}=P(T^{k} (x_{0})<T^{k-1} (x_{0}))$ in step $ k $ of a Collataz procedure…
A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…
The Collatz problem with $3x+k$ is revisited. Positive and negative limit cycles are given up to k=9997 starting with $x_0=-2\cdot10^7...+2\cdot10^7$. A simple relation between the probability distribution for the Syracuse iterates for…
Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the…
In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…
Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics…
We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly…
Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…
In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear…
In this paper, we study tree--like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of…
We construct a family of trees with independence numbers going to infinity for which the log-concavity relation for the independent set sequence of a tree $T$ in the family fails at around $\alpha(T)\left(1-1/(16\log \alpha(T))\right)$.…
A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special…
The Collatz graph is a directed graph with natural number nodes and where there is an edge from node $x$ to node $T(x)=T_0(x)=x/2$ if $x$ is even, or to node $T(x)=T_1(x)=\frac{3x+1}{2}$ if $x$ is odd. Studying the Collatz graph in binary…
We consider the following question: Which real sequences (a(n)) that satisfy a linear recurrence with constant coefficients are positive for sufficiently large n? We show that the answer is negative for both (a(n)) and (-a(n)), if the…
We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for…
The Collatz variations pattern seems not to have any recurrence relation between numbers. But knowing that there is at least a natural number that converges after several iterations we construct a function $f_{X,Y}$ that is equal to the…
This paper presents some considerations about the Goldbach's conjecture (GC). The work is based on elementary results of the number theory and it provides a constructive method that permits, given an even integer, to find at least a pair of…
The Collatz conjecture (also known as the $3x+1$ problem) concerns the behavior of the discrete dynamical system on the positive integers defined by iteration of the so-called $3x + 1$ function. We investigate analogous dynamical systems in…
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of…
A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. For such sets, we demonstrate the existence of a sequence of algebraic…