Number Theory

Flows of the Riemann Hypothesis

Authors: Tai-Choon Yoon, Yina Yoon
Comments: 4 Pages.

The Riemann hypothesis is a mathematical conjecture that relates to the calculation of prime numbers through the Riemann product formula, which represents the product of Riemann zeta function and factorial. There were flows in deriving ∫x^(s-1)/(e^x-1) dx from Riemann product formula and, in attempting to represent the negative region by substituting x with —x. Furthermore, asserting that the Riemann zeta function, in the absence of a definition for negative factorial, obtains trivial zeros for negative even numbers through the Bernoulli exponential generating formula in the negative domain is also incorrect.
Category: Number Theory

About an Integral in Valean's Book

Authors: Edgar Valdebenito
Comments: 3 Pages.

The evaluation of integrals is an important subject in mathematics, physics and applied sciences. In this note we give some integrals fot pi^3 .
Category: Number Theory

The Geometric Collatz Correspondence

Authors: Darcy Thomas
Comments: 28 Pages. This is a final, and more complete, version of the paper. It is better categorized number theory.

The Collatz Conjecture, one of the most renowned unsolved problems in mathematics, presents adeceptive simplicity that has perplexed both experts and novices. Distinctive in nature, it leaves manyunsure of how to approach its analysis. My exploration into this enigma has unveiled two compellingconnections: firstly, a link between Collatz orbits and Pythagorean Triples; secondly, a tie to theproblem of tiling a 2D plane. This latter association suggests a potential relationship with PenroseTilings, which are notable for their non-repetitive plane tiling. This quality, reminiscent of theunpredictable yet non-repeating trajectories of Collatz sequences, provides a novel avenue to probethe conjecture’s complexities. To clarify these connections, I introduce a framework that interpretsthe Collatz Function as a process that maps each integer to a unique point on the complex plane.In a curious twist, my exploration into the 3D geometric interpretation of the Collatz Function has nudged open a small, yet intriguing door to a potential parallel in the world of physics. A subtle link appears to manifest between the properties of certain objects in this space and the atomic energy spectral series of hydrogen, a fundamental aspect in quantum mechanics. While this connection is in its early stages and the depth of its significance is yet to be fully unveiled, it subtly implies a simple merging where pure mathematics and applied physics might come together.The findings in this paper have led me to pursue development of a new type of number I call a Cam number, which stands for "complex and massive", indicating that it is a number with properties that on one hand act like a scalar, but on the other hand act as a complex number. Cam numbers can be thought of as having somewhat dual identities which reveal their properties and behavior under iterations of the Collatz Function. This paper serves as a motivator for a pursuit of a theory of Cam numbers.
Category: Number Theory

Sum of Three Cubes Explored - Proof

Authors: James DeCoste
Comments: 15 Pages. Contact: jbdecoste@eastlink.ca

Using already known techniques along with some not so obvious innovations on my part, I was able to show (prove) that there are solutions for all K (except those of the form 9m+/-4 and 9m+/-5 which are impossible) for +/-K = +/- (x^3) +/- (y^3) +/- (z^3). A further stipulation is that x, y and z must be whole numbers that can be a combination of positives and negatives. This is achieved through simple subtraction. Setting up a table showing that all K can be represented using a multiple of 27 plus a mask lends validity to a portion of the proof. These representations may and often do contain many more than the required number of cubes summed up. I side step that problem by showing that no matter the K picked and how ever many cubes are required to create it in my representations, they can all be reduced to a maximum of cubes summed. Exactly what we require for the proof. Having done that we are complete. The three new cubes we have just reduced to are already included in table. They are items I have already represented in the above format.
Category: Number Theory

A Direct, Simple, and Basic Computation of a Difference of Two Dilogarithms

Authors: Hervé Gandran-Tomeng
Comments: 2 Pages.

The computation of dilog(sqrt(2)-1)-dilog(1-sqrt(2)) is performed.
Category: Number Theory

Collatz Conjecture Proof for Special Integer Subsets and a Unified Criterion for Twin Prime Identification

Authors: Budee U. Zaman
Comments: 5 Pages.

This paper presents a proof of the Collatz conjecture for a specific subset of positive integers, those formed by multiplying a prime number "p" greater than three with an odd integer "u" derived using Fermat’s little theorem. Additionally, we introduce a novel screening criterion for identifying candidate twin primes, extending our previous work linking twin primes (p and p+2) with the equation 2(p−2) = pu+v, where unique solutions for u and v are required. This unified criterion offers a promising approach to twin prime identification within a wider range of integers, further advancing research in this mathematical domain.
Category: Number Theory

The Floor and Ceiling Functions

Authors: Edgar Valdebenito
Comments: 4 Pages.

In this note we give some properties of the Floor and Ceiling functions.
Category: Number Theory

Proof of the Collatz Conjecture, 3x+1 by Adriano Bertaggia

Authors: Adriano Bertaggia
Comments: 81 Pages.

We will proof that the 3x+1 conjecture is true, using modular arithmetic and a new approach based on an ancient symbol THE ENNEAGRAMMA. We will show that for every integer n, n ≡ 1 (mod 2) if and only if 3n+1 ≡ 4 (mod 6). With the help of directed graphs, flow and block diagrams we will find 1 equation which, applying the 2 conditions, links all the odd numbers and consequently the positive integers to the powers of 2. We will find the analytical equation of the function. We will show how "numerical gravity" arises from the deterministic divisibility that the combinations of integers allow. We will go up the Collatz graph represented by the inverse function which forms a tree with the exception of the cycle 1-4-2-1... We will show how all positive integers are present in the tree, that is connected to the number 1, making extensive use of graphs, tables and colors to represent the beauty of mathematics. We will follow the exact chronology of the insights. Careful observation of the numbers will return an elementary (-a)rithmetic (double logical negation equals affirmation). We will not omit steps that are obvious, since these are the substrate on which the approach is based. We hope you can appreciate the extreme simplicity, harmony and rhythm that the numbers manifest.
Category: Number Theory

Authors: G. Hervé
Comments: 2 Pages. (Abstract added by viXra Admin)

[This note is about a proof of a conjecture on the Riemann zeta-function at even integers]
Category: Number Theory

Expressing Even Numbers Beyond 6 as Sums of Two Primes

Authors: Budee U. Zaman
Comments: 3 Pages.

The "strong Goldbach conjecture" posits that any even number exceeding 6 can be represented as the sum of two prime numbers. This study explores this hypothesis, leveraging the constancy of odd integer quantities and cumulative sums within positive integers. By identifying odd prime numbers, pα1and pα2, within [3, n] and (n, 2n-2) intervals, we demonstrate a transformative process grounded in the unchanging nature of odd number counts and their cumulative sums. Through this process, we establish the equation 2n =pα1 + pα2, offering a significant stride in unraveling the enigmatic core of the strong Goldbach conjecture.
Category: Number Theory

The Sum of Positive and Negative Prime Numbers are Equal

Authors: Budee U. Zaman
Comments: 8 Pages.

This paper unveils a profound equation that harnesses the power of natural numbers to establish a captivating theorem: the balance between positive and negative prime numbers’ summation, intricately linked through the medium of natural numbers. As a corollary, the essence ofnatural numbers emerges as a testament to the harmonious interplay between even and odd elements. Notably, we expose the remarkable revelation that odd numbers find expression as both the aggregate of prime divisors and the sum of prime numbers, fusing diverse mathematical concepts into an elegant unity. This work reshapes the landscape of number theory, illuminating the hidden connections between primes, naturals, andtheir arithmetical compositions.
Category: Number Theory

Double Inequalities Related to Approximate Formulas of Euler-Mascheroni Constant with Continuedfraction

Authors: JiSong Ro, SongIl Kang, JinSong Yu, HyonChol Kim
Comments: 6 Pages.

In this paper, we present some new double inequalities starting from the approximate formula for Euler-Mascheroni constant the newly obtained by us.
Category: Number Theory

New Approaches to Riemann Hypothesis Solution

Authors: Miroslav Sukenik, Magdaléna Súkeníková
Comments: 3 Pages.

In the article, we assume that the Golden Ratio plays a fundamental role in alocalosation of non-trivial zero points of the Riemann Zeta function on the critical line s = 0.5.
Category: Number Theory

Proof by Analysis-Synthesis and by the Absurd of Goldbach's Conjecture

Authors: Yves Désiré Ipolo
Comments: 8 Pages. In French

This article proposes an original approach never before addressed to demonstrate the Goldbach conjecture which uses reasoning by analysis-synthesis followed by reasoning by the absurd. I would like to put forward the key to the demonstration in the form of a revisited Goldbach Conjecture 1: "For every natural integer n strictly greater than 3, there exists at least one prime natural integer p which is prime with n such that 2n-p is prime and prime with n." The "Goldbach solutions", if they exist, are necessarily prime with 2n.
Category: Number Theory

Hilbert and Pólya Conjecture, Dynamical System, Prime Numbers, Black Holes, Quantum Mechanics, and the Riemann Hypothesis

Authors: Mohamed sghiar
Comments: 13 Pages.

In mathematics, the search for exact formulas giving all the prime numbers, certain families of prime numbers or the n-th prime number has generally proved to be vain, which has led to contenting oneself with approximate formulas [8]. The purpose of this article is to give a simple function to produce the list of all prime numbers.And then I give a generalization of this result and we show a link with the quantum mechanics and the attraction of black Holes. And I give a new proof of lemma 1 which gave a proof of the Riemann hypothesis [4]. Finally another excellent new proof o f the Riemann hypothesisis given and I deduce the proof of Hilbert Polya's conjecture
Category: Number Theory

A Formula for Mertens' Function and Its Applications

Authors: Roy L. Lewis Jr.
Comments: 12 Pages.

In this article, we prove the limit formula lim M(x) / &pi(x) = lim h / log(x) = 0, h = a constant for Mertens' function M(x) using arithmetic and analytic arguments based on theorems for the prime counting function &pi(x) and the series &sum &mu(k)/k. The formula is evaluated using limit theorems to give: an alternative proof of lim M(x)/ x = 0, a new disproof of Mertens' conjecture, proof of the Odlyzko--te Riele conjecture and a disproof of the Riemann hypothesis based on Littlewood's equivalence theorem.
Category: Number Theory

Proof of the Odd Goldbach's Conjecture

Authors: Samuel Ferrer
Comments: 2 Pages. (Abstract added to Article by viXra Admin - Please conform!)

In this work a proof is presented provided that the original Goldbach’s conjecture(Goldbach’s) has been verified.
Category: Number Theory