Related papers: A Refinement of the $3x+1$ Conjecture
In this paper, we provide a comprehensive solution to the open problem regarding the existence of a recurrence formula for computing fixed points of the Josephus function precisely when the reduction constant is three. Incorporating this…
Conjecturing formulas and other symbolic relations occurs frequently in number theory and combinatorics. If we could automate conjecturing, we could benefit not only from speeding up, but also from finding conjectures previously out of our…
We investigate the probability that a random odd composite number passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to $2^{200}$, our results improve on prior…
We consider a family of piecewise contractions admitting a rotation number and defined for every $x\in[0,1)$ by $f(x)=\lambda x + \delta + d \theta_a(x) \pmod 1$, where $\lambda\in(0,1)$, $d\in(0,1-\lambda)$, $\delta\in[0,1]$, $a\in[0,1]$…
Let $a,b$ and $n$ be positive integers with $a>b$. In this note, we prove that $$(2bn+1)(2bn+3){2bn \choose bn}\bigg|3(a-b)(3a-b){2an \choose an}{an\choose bn}.$$ This confirms a recent conjecture of Amdeberhan and Moll.
Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a…
This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erd\H{o}s-Mirksy conjecture which states that there are infinitely many positive integers…
The scope of the present work is to explain why it is true that all N have a distinct position in The Collatz Tree (The Collatz Graph)
Large language models (LLMs) are now used in multi-turn workflows, but we still lack a clear way to measure when iteration helps and when it hurts. We present an evaluation framework for iterative refinement that spans ideation, code, and…
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…
In this article we present set of infinite natural numbers which satisfies the conjecture $3n+1$.
We introduce an alternative approach for constrained mathematical programming problems. It rests on two main aspects: an efficient way to compute optimal solutions for unconstrained problems, and multipliers regarded as variables for a…
It is well known that the Collatz Conjecture can be reinterpreted as the Collatz Graph with root vertex 1, asking whether all positive integers are within the tree generated. It is further known that any cycle in the Collatz Graph can be…
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…
Numerical relativity is finally approaching a state where the evolution of rather general (3+1)-dimensional data sets can be computed in order to solve the Einstein equations. After a general introduction, three topics of current interest…
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…
Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum…
I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…
We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling…
In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved:…