Related papers: A Fractional $3n+1$ Conjecture

An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.

R. L. Graham and H. O. Pollak observed that the sequence $$u_1=1,\qquad u_{n+1}=\lfloor \sqrt{2} (u_n+1/2)\rfloor, \quad n\geq 1,$$ has the curious property that the sequence of numbers $(u_{2n+1}-2u_{2n-1})_{n\geq 1}$ denotes the binary…

Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and…

We consider the following conjecture (from Huang, et al): Let $\Delta^+$ denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic…

In this article, we define a very important sequence of functions, all the functions of this sequence present behaviors very close to that of the Collatz function. The study of such functions allows us to obtain very interesting results…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

The Collatz problem is one of many names (the Collatz Problem, the Syracuse Problem, the Hailstone Problem, the 3x+1 problem). Most commonly, however, the problem goes by either the 3x+1 problem or the Collatz problem. In addition to having…

Sequence of numbers generated by the recurrence relation based on the Collatz conjecture is investigated. An arithmetic operation on the Collatz conjecture is called descending operation, and ascending operation is carried out reversely to…

The Collatz problem is generalized into $3n + 3^k$ problem. It is shown that as long as the Collatz function iterates converge to the cycle passing through the number 1, the $3n + 3^k$ sequence converges to the cycle passing through the…

The objective of this paper is to show (a)=(b)=(c) as rational functions of $T$, $U$ for (a), (b), (c) given by (a) continued fractions of length $2^{n+1}-1$ with explicit partial denominators in $\left\{-T,U^{-1}T\right\}$, (b) truncated…

Considering all possible paths that a natural number can take following the rules of the algorithm proposed in the Collatz conjecture we construct a graph that can be interpreted as an infinite network that contemplates all possible paths…

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most…

Given a strictly increasing sequence of positive real numbers tending to infinity $(q_{n})_{n=1}^{\infty}$, and an arbitrary sequence of real numbers $(r_{n})_{n=1}^{\infty}.$ We study the set of $\alpha\in(1,\infty)$ for which…

Let $T(n)=\left\{\begin{array}{ll}3n+1&(n\hbox{ odd})\frac n2&(n\hbox{ even})\end{array}\right.$ ($n\in\mathbb Z$). We call "the orbit of the integer $n$", the set $$ \mathcal O_n:=\{m\in\mathbb Z\;:\;\exists k\ge0,\ m=T^k(n)\} $$ and we…

This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration $T(n) = (3n+1)/2$ for $n$ odd, $T(n) = n/2$ for $n$ even. Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias…

We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$…

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…

Let $\varepsilon>0$. We construct an explicit, full-measure set of $\alpha \in[0,1]$ such that if $\gamma \in \mathbb{R}$ then, for almost all $\beta \in[0,1]$, if $\delta \in \mathbb{R}$ then there are infinitely many integers $n\geq 1$…

The Collatz conjecture states that repeated steps of $n\mathrm{\to }\mathrm{3}n\mathrm{+1}$ at odd numbers and $n\mathrm{\to }n\mathrm{/2}$ at even numbers amount to walks over root paths to the branching number $c=4$ in the `trivial'…