Related papers: The Proof of the Collatz Conjecture
Paul Erdos claimed that mathematics is not yet ready to settle the 3x+1 conjecture. I agree, but very soon it will be! With the exponential growth of computer-generated mathematics, we (or rather our silicon brethrern) would have a shot at…
In this article we present set of infinite natural numbers which satisfies the conjecture $3n+1$.
In the paper, from the point of view of recurrent numbers of the Jacobsthal type, the Collatz problem with the general aq+-1 function of conjecture odd positive integers q from the set of natural numbers is investigated. Formulated…
The famous 3x + 1 problem of L. Collatz needs no introduction; however, this paper concerns a lesser-known, but similarly unresolved, precursor problem : the Original Collatz Conjecture, or OCC. We demonstrate that the core arithmetic…
We developed an algorithm that easily goes from one odd number to the next odd number in binary representation for the reduced forward Collatz map (Syracuse function). The algorithm indicates when an odd number can grow or shrink to the…
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…
In this paper we introduce and discuss the sequence of \emph{real numbers} defined as $u_0 \in \mathbb R$ and $u_{n+1} = \Delta(u_n)$ where \begin{equation*} \Delta(x) = \begin{cases} \frac{x}{2} &\text{if }…
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…
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.
In this paper, we show that if the numbers in the range $[1,2^n]$ satisfy Collatz conjecture, then almost all integers in the range $[2^n+1,2^{n+1}]$ will satisfy the conjecture as $n \to \infty$. The previous statement is equivalent to…
An operational approach to the Collatz Conjecture is presented. Scenarios are defined as strings of characters "s" (for "spike") and "d" (for "down") which symbolize the Collatz operations (3m+1)/2 and m/2 in a Collatz Series connecting two…
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…
Pairs of consecutive integers have the same height in the Collatz problem with surprising frequency. Garner gave a conjectural family of conditions for exactly when this occurs. Our main result is an infinite family of counterexamples to…
It is shown that every Collatz sequence $C(s)$ consists only of same structured finite subsequences $C^h(s)$ for $s\equiv9\ (mod\ 12)$ or $C^t(s)$ for $s\equiv3,7\ (mod\ 12)$. For starting numbers of specific residue classes ($mod\…
The Collatz problem is related to the fixed point problem, and is widely used in mathematics. It has attracted a wide range of math enthusiasts, but is still difficult to solve. So, this article aimed to study the extension of the Collatz…
Although 10^230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n >= 3} is defined as follows. Start with n, and…
This paper is a numerical evaluation of some trajectories of the Collatz function. Specifically, I assess the coalescence points of each integer $n\equiv 0 (\bmod{2})$ and $n\equiv 2(\bmod{3})$ through a sophisticated algorithm that has…
In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new…
We present a comprehensive structural analysis of the Collatz conjecture through ~1014 computational experiments yielding 630 formal results. By systematically deploying 29 distinct mathematical paradigms--including transfer operator…
Given a finite nonempty sequence S of integers, write it as XY^k, where Y^k is a power of greatest exponent that is a suffix of S: this k is the curling number of S. The Curling Number Conjecture is that if one starts with any initial…