Supplement to the Collatz Conjecture

For any natural number was created the supplement sequence, that is convergent together with the original sequence. The parameter - index was defined, that is the same tor both sequences. This new method provides the following results: All natural numbers were distributed into different classes according to the corresponding indexes; The analytic formulas ( not by computer performed routine calculations) were produced, the formulas for groups of consecutive natural numbers of different lengths, having the same index; The new algorithm to find index for any natural number was constructed and proved

A simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the 3x+13x+1 problem

This paper gives a simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the $3x+1$ problem (see \cite{Wirsching} and \cite{Goodwin}). This representation permits to compute all the ascending Collatz sequences $(f^{(i)}(n),\: i=1,b-1)$ with a last value $f^{(b)}(n)=1.$ Other periodic sequences connected to $1$ are also identified

Predictable trajectories of the reduced Collatz iteration and a possible pathway to the proof of the Collatz conjecture (Version 2)

I show here that there are three different kinds of iterations for the reduced Collatz algorithm; depending on whether the root of the number is odd or even. There is only one kind of iteration if the root is odd and two kinds if the root is even. I also show that iterations on numbers with odd roots will cause an increase in value and eventually lead to an even rooted number. The iterations on even rooted numbers will subsequently cause a decrease in values. Because increase in values during the odd root iterations are bounded, I conclude that the Collatz iteration cannot veer to infinity. Since the sequence generated by the Collatz iteration is infinite and the values of the numbers do not veer to infinity it must either cycle and/or converge. I postulate that any cycling must occur with alternating types of iterations: e.g. odd rooted iterations which cause the values of the numbers to increase followed by even rooted iterations which causes the values to decrease. I show here that for simpler types of cycles, valid values of odd rooted or even rooted numbers are only found in a narrow gap which closes as the number of iterations increase. I further generalize to all types of odd-even and even-odd iterations. Given that previous work has shown that only very large non-trivial cycles are feasible during the Collatz iteration and this study shows the low probability of large simple cycles, leads us to the conclusion most likely cycles other than the trivial cycle are not possible during the Collatz iteration

New Experimental Investigations for the 3x+1 Problem: The Binary Projection of the Collatz Map

The 3x + 1 Problem, or the Collatz Conjecture, was originally developed in the early 1930\u27s. It has remained unsolved for over eighty years. Throughout its history, traditional methods of mathematical problem solving have only succeeded in proving heuristic properties of the mapping. Because the problem has proven to be so difficult to solve, many think it might be undecidable. In this paper we brie y follow the history of the 3x + 1 problem from its creation in the 1930\u27s to the modern day. Its history is tied into the development of the Cosper Algorithm, which maps binary sequences into integer families. The Then we discuss the pseudo-code which the Cosper Algorithm is based upon. A simple example is provided to demonstrate the Cosper Algorithm. Afterwards, the generalized 3x + k problem is considered yielding two definitions: k-dependent and k-independent cycles. Finally, some images are provided of various k-dependent cycles

Computable Randomness, and Coding the Orbits of the Collatz Map

In this thesis I will look at a definition of computable randomness from Algorithmic Information Theory as defined by Andre Nies through the lens of Computable Analaysis asdefined by Klaus Weihrauch. I will show that despite the fact that these two paradigmsgenerate distinct classes of computable supermartingales, the class of sets on which nocomputable supermartingale succeeds of either type is identical. Therefore, both theoriesgenerate the same collection of computably random sets. I will then consider how onemight apply some of the techniques in Algorithmic Information Theory, including prefixfree codes and the Kraft Inequality, to the study of the Collatz Conjecture.Keywords: Collatz Conjecture, Computability, Algorithmic Randomness, Computable Analysis, Complexit