Related papers: Working with 2s and 3s
The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd,…
The purpose of this study is to show how to get a necessary criterion for prime numbers with the help of special matrices. My special interest lies in the empirical research of these matrices and their patterns, structures and symmetries.…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
The Collatz conjecture (also known as the $3x+1$ problem) concerns the behavior of the discrete dynamical system on the positive integers defined by iteration of the so-called $3x + 1$ function. We investigate analogous dynamical systems in…
A structured approach for the Collatz conjecture is presented using just the odd integers that are, in turn, divided into categories based on the roles they play such as Starter, Intermediary and Terminal. The expression 4x+1 is used as a…
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 present work focuses on the study of the renowned Collatz conjecture, also known as the $3x +1$ problem. The distinguished analysis approach lies on the dynamics of an iterative map in binary form. A new estimation of the enlargement of…
We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…
In this paper, we discuss the well known 3x+1 conjecture in form of the accelerated Collatz function T defined on the positive odd integers. We present a sequence of quotient spaces and an invertible map that are intrinsically related to…
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 can be stated as: using the reduced Collatz function $C(n) = (3n+1)/2^x$ where $2^x$ is the largest power of 2 that divides $3n+1$, any odd integer $n$ will eventually reach 1 in $j$ iterations such that $C^j(n) = 1$.…
The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…
According to some discussions based on syllogism, we present results on the binary Goldbach conjecture in three categories: results that are weaker than the Goldbach conjecture, sufficient conditions for the Goldbach conjecture, and results…
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…
For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that…
The counting function for the numbers satisfying the Collatz conjecture is studied. A related exponential congruence equation is investigated, yielding a method to construct its solutions from free variables, and enabling us to find at…
On the set of positive integers, we consider the iterative process that maps $n$ to either $\frac{3n+1}{2}$ or $\frac{n}{2}$ depending on the parity of $n$. The Collatz conjecture states that all such sequences eventually enter the trivial…
The Collatz function is defined as C(n) = n / 2 if n is even and C(n) = 3n + 1 if n is odd. The Collatz conjecture states that every sequence generated by the Collatz function ends with the cycle (4, 2, 1) after a finite number of…
In this paper, we investigate a class of Collatz-like problems associated with weakly and strongly admissible triplets of integers. This framework extends the classical Collatz mapping, providing a systematic method for generating triplets…
In this article, we give two different proofs of why the Collatz Conjecture is false.