Related papers: An Exploration Into the Collatz Conjecture with Ch…
Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to…
Circular permutations on {1,2,...,n} that avoid a given pattern correspond to ordinary (linear) permutations that end with n and avoid all cyclic rotations of the pattern. Three letter patterns are all but unavoidable in circular…
It has been conjectured that the sequence $(3/2)^n$ modulo $1$ is uniformly distributed. The distribution of this sequence is signifcant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we…
Let $T \colon \mathbb{N} \to \mathbb{N}$ denote the $3x+1$ function, where $T(n)=n/2$ if $n$ is even, $T(n)=(3n+1)/2$ if $n$ is odd. As an accelerated version of $T$, we define a jump at $n \ge 1$ by jp$(n) = T^{(\ell)}(n)$, where $\ell$ is…
In this paper we show the probabilistic convergence of the original Collatz (3n + 1) (or Hotpo) sequence to unity. A generalized form of the Collatz sequence (GCS) is proposed subsequently. Unlike Hotpo, an instance of a GCS can converge to…
We investigate the structure of Collatz path sequences $\{F^k(n)\}_{k=0}^{\infty}$ for positive integers $n$, where $F$ denotes the standard Collatz map. By classifying natural numbers into residue classes modulo~4, we establish that the…
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…
This paper introduces prime holdout problems, a problem class related to the Collatz conjecture. After applying a linear function, instead of removing a finite set of prime factors, a holdout problem specifies a set of primes to be…
This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all…
We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern…
The 3x+1 problem concerns iteration of the map T(n) =(3n+1)/2 if n odd; n/2 if n even. The 3x +1 Conjecture asserts that for every positive integer n>1 the forward orbit of n includes the integer 1. This paper is an annotated bibliography…
A multipermutation with $k$ copies each of $1\ldots n$ is Carlitz if neighbours are different. We enumerate these objects for $k=2,3,4$ and derive recurrences. In particular, we prove and improve a conjectured recurrence for $k=3$, stated…
A recast of the standard residue-class analysis of the 3x+1 (Collatz) map in terms of two elementary operators on arithmetic progressions. The resulting calculus (i) splits any progression into its even and odd subsequences in a single…
We give new Turing machines that simulate the iteration of the Collatz 3x+1 function. First, a never halting Turing machine with 3 states and 4 symbols, improving the known 3x5 and 4x4 Turing machines. Second, Turing machines that halt on…
The $3x+1$ Problem asks if whether for every natural number $n$, there exists a finite number of iterations of the piecewise function $$f(2n)=n, \quad f(2n-1)=6n-2, $$ with an iterate equal to the number $1$, or in other words, every…
In the Collatz 3x+1 problem, there are 3 possibilities: Starting from any positive number, we either reach the trivial loop (1,4,2), end up in a non-trivial loop, or go until infinity. In this paper, we shall show that if a non-trivial loop…
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…
Following up on earlier work, I suggest why there are no mappings to infinity under the Collatz conjecture, nor under other mappings of the generalization $3n+p$, where $p$ is odd.
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…
We present a formulation of the Collatz conjecture that is potentially more amenable to modeling and analysis by automated termination checking tools.