Related papers: Two different scenarios when the Collatz Conjectur…
For a long time, Collatz Conjecture has been assumed to be true, although a formal proof has eluded all efforts to date. In this article, evidence is presented that suggests such an assumption is incorrect. By analysing the stopping times…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatz's theorem cannot be proved in any…
This paper studies the proof of Collatz conjecture for some set of sequence of odd numbers with infinite number of elements. These set generalized to the set which contains all positive odd integers. This extension assumed to be the proof…
In this paper, the abc conjecture is negated under certain conditions
This paper explores special conditions on the starting value of a Collatz sequence which imply that the Collatz conjecture is true. This is the result of the collaboration of a retired mathematics professor (Koelzer) and a retired physics…
In this paper, we show that any proof of the Collatz 3n+1 Conjecture must have an infinite number of lines; therefore, no formal proof is possible.
Some simple facts are proved ruling the Collatz tree and the chains of vertices appearing in it, leading to the reduction of the number of significant elements appearing in the tree. Although the Collatz conjecture remains open, these fact…
The Collatz conjecture is a famous math problem that was introduced by Lothar Collatz in 1937, and nobody has yet succeeded in proving or disproving it. In this article, I will analyze this problem with a new approach and I will discuss my…
We build a variant of Collatz Conjecture for polynomials over $\mathbb{F}_2$ and we prove that it is solved. By the way, we give several examples.
In this research, an optimal algorithm for the Collatz conjecture is presented. Properties such as the convergence of the algorithm and an equation that relates the algorithm to the classical Collatz conjecture are obtained. It is validated…
In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz…
We study a natural analogue of Collatz's Conjecture for polynomials over $\mathbb{F}_2$.
The Collatz conjecture implies that an iterated function sequence under a certain linear operator, beginning with a certain complex valued function, must converge to a certain complex function.
An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.
Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the…
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…
Collatz Conjecture sequences increase and decrease in seemingly random fashion. By identifying and analyzing the forms of numbers, we discover that Collatz sequences are governed by very specific, well-defined rules, which we call cascades.
It is well known that the following Collatz Conjecture is one of the unsolved problems in mathematics. Collatz Conjecture: For any positive integer $n>1$, the following recursive algorithm will convergent to 1 by a finite number of steps.…
We provide a proof and a counterexample to two conjectures made by N. Kuznetsov.