Characterization of Syracuse Maps as Non-singular Power-Bounded Transformations and the Inverse Image

In this thesis, we will introduce the well-known Collatz Conjecture, discussing some of the previous work done on this problem. Next, we give a dynamical system characterization of an arbitrary map V : N → N, which applies to both the Collatz Map and the related Syracuse maps. In the third section of this thesis, we discuss some number theoretic properties of the Collatz inverse image, comparing some properties to one particular subclass of the Syracuse maps. In the last section, we analyze previous density results, looking at what this does and does not give in terms of density results on the elements with unbounded Collatz trajectories, denoted by D2. We also motivate the study of density results regarding D2, by showing how previous results may be strengthened with the assumption that D2 has asymptoptic density 0. Part of the results in this thesis can be found in [2].Bachelor of Scienc

Problems in number theory from busy beaver competition

By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the following one: How can a small Turing machine manage to produce very big numbers? It provides the following answer: mostly by simulating Collatz-like functions, that are generalizations of the famous 3x+1 function. These functions, like the 3x+1 function, lead to new unsolved problems in number theory.Comment: 35 page

A novel theoretical framework formulated for information discovery from number system and Collatz conjecture data

Newly discovered fundamental theories (metamathematics) of integer numbers may be used to formalise and formulate a new theoretical number system from which other formal analytical frameworks may be discovered, primed and developed. The proposed number system, as well as its most general framework which is based on the modelling results derived from an investigation of the Collatz conjecture (i.e., the 3x+1 problem), has emerged as an effective exploratory tool for visualising, mining and extracting new knowledge about quite a number of mathematical theorems and conjectures, including the Collatz conjecture. Here, we introduce and demonstrate many known applications of this prime framework and show the subsequent results of further analyses as new evidences to justify the claimed fascinating capabilities of the proposed framework in computational mathematics, including number theory and discrete mathematics