arXiv subjects

Raouf Rajab

Publications and source records attributed to Raouf Rajab.

5 recordsLinked to original sources

arXiv2022

Characteristic numbers and characteristic equations of parity vectors of Collatz sequences

The present work deals with the characterization of parity vectors of Collatz sequences (of finite and infinite length). Such a characterization leads to the determination of several numbers (integers or non-integers) that we call the characteristic numbers of a given parity vector. Some characteristic numbers are linked together by equations that can be called characteristic equations of the considered parity vector. If a parity vector v of finite length n contains the first n terms of a parity vector V of infinite length then all the characteristic numbers of v are considered as characteristic numbers of order n of the infinite vector V. The limits of nth order characteristic numbers when n tends to infinity constitute the absolute characteristic numbers of the infinite vector V and they allow determining its behavior and properties. In this paper, we study the properties of parity vectors of infinite length based on their characteristic numbers. Then, we establish the relations between the first term of a Collatz sequence and its parity vector. Finally, still based on the characteristic numbers, we determine some conditions of existence and non-existence of divergent sequences.

math.GM↗

arXiv2021

The sequence of Collatz functions, exceptionality of the 3n+1 function and the notion of Collatz generalized Matrix

In this article, we define a very important sequence of functions, all the functions of this sequence present behaviors very close to that of the Collatz function. The study of such functions allows us to obtain very interesting results concerning the collatz conjecture and more precisely concerning the generalization of this problem, the nature of the Collatz function, on the statement of the Collatz conjecture we show that this conjecture can be stated for each function on a well-determined subset of Z and other interesting properties in relation to the behaviors of Collatz sequences are studied and proved in this article.

math.GM↗

arXiv2021

Classification of Collatz infinite sequences

In the present paper, we are interested in classifying of Collatz sequences on based to the different behavior of these sequences when their lengths tend to infinity. A Collatz infinite sequence can be defined as an infinite ordered set of positive integers such that the term of rank n is results of applying Collatz map n times to the first term. Such term can be expressed on the form Tn(P)=A(P,n)P+B(P,n). When n tends to infinity, each function among the two partial coefficients denoted by A(P,n) and B(P,n) behaves in different ways. This allows us to determine all categories of Collatz infinite sequences. First, we carry out a classification of Collatz infinite sequences on based of the different possible limits of the two coefficients. In second time, we determine the different proportions of every class of the infinite sequences. Note that results obtained do not represent a proof of a Collatz conjecture but they have a strong relationship with this conjecture and it allows us to better understand the behavior of Collatz sequences when n tend to infinity.

math.GM↗

arXiv2021

General formulas of global characteristic coefficients of Collatz function

The purpose of this paper is to show three general formulas of three global characteristic coefficients of Collatz function. The Collatz function is defined by the following operation on an arbitrary positive integer if N is odd multiply it by 3 and add 1 then the sum obtained is divided by 2, if N is even divide it by 2. Based on the principle, we define the n-order function denoted by T^n such as the different expressions of that function are results of applying Collatz function n times to a natural numbers which are expressed in well determined forms. Based on these expressions, we can characterize that n-order function by three global characteristic coefficients. In the first, we define these three global coefficients. Secondly, we show that each global characteristic coefficient has a general expression as a function of n.

math.NT↗

arXiv2021

The fundamental properties characterizing the structural behaviors of Collatz sequences

This work represents an in-depth study of the structural behavior of the Collatz sequences. We consider a finite arithmetic progression with a common difference is 2 and the number of terms in the sequence is equal to 2^n . After, we consider a 2^n x(n+1) matrix ((n+1) columns and 2^n rows) such as the first column contains the terms of arithmetic progression and each row of the matrix represent a finite Collatz sequence. Then, each element of the matrix will be replaced by 0 or 1 according to the following rule: the even integer is replaced by 0 and the odd integer is replaced by 1. We obtained a table contains all binary permutation with repetition this property is called the property of structural complementarily. Based on these tables, we can determine any other fundamentals properties characterizing the behavior of Collatz sequences. Thus, we can distinguish two other important properties such as the property of structural cyclical behavior characterized by a structural periodicity and the structural sliding property that can be called also the property of structural divergence.

math.GM↗