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[27] viXra:2112.0166 [pdf] submitted on 2021-12-30 23:14:57
Authors: Gregory M. Sobko
Comments: 89 Pages.
I suggest a probabilistic approach that helps to address some classical questions and problems of Number Theory, like the Goldbach Conjecture [1], distributions of twin- and d-primes, prime numbers among arithmetic sequences and others.
The concepts of ‘randomness’ and ‘independence’ relevant to number-theoretic problems are discussed here, and the basic concepts of divisibility of natural number are interpreted in terms
of probability spaces and appropriate probability distributions on classes of congruence.
I analyze and demonstrate the importance of Zeta probability distribution and prove theorems stating the equivalence of probabilistic independence of divisibility of random integers by coprime factors, and the fact that random variables with the property of independence of coprime factors must have Zeta probability distribution. The idea to use Zeta distribution is motivated by the fact that it provides the validity of the probabilistic Cramér’s model for asymptotic prime number distribution, in full agreement with the Prime Number Theorem. Multiplicative and additive models with recurrent equations for generating sequences of prime numbers are derived based on the reduced Sieve of Eratosthenes Algorithm. This allows to interpret such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by sets of all prime numbers, representing paths of stochastic dynamical systems.
The H. Cramér’s model for probability distribution of primes is modified as a generalized predictable non-stationary Bernoulli process with unequally distributed terms that are asymptotically pairwise independent. This model is applied to analyze the sequences of primes generated by appropriate random walks. With intense use of Zeta probability distribution, it seems possible by using the modified Cramér’s model to approximate probability distribution of various arithmetic function.
Since probabilistic approach meets certain skepticism and even disbelief from a part
of mathematicians working in traditional manner in Number Theory, I decided to attack
the problem of Strong Goldbach Conjecture (SGC) from pure deterministic point of view.
As a result, I derived a recursive formula which generates a sequence {G(m)} of consecutive nonempty Goldbach sets. Each Goldbach set G(m) is asset of all prime numbers solving equation p + p’ = 2m for any natural number m > 2. The recursive formula justifies SGC by mathematical induction. Thus, this work represents two independent proofs of validity for Strong Goldbach Conjecture.
Category: Number Theory
[26] viXra:2112.0165 [pdf] submitted on 2021-12-31 00:14:30
Authors: Marco Ripà, Luca Onnis
Comments: 7 Pages.
In the present paper we provide a general formula which let us easily calculate the number of stable digits of any integer tetration base a∈ℕU{0}. The number of stable digits, at the given height of the power tower, indicates how many of the last digits of the (generic) tetration are frozen. Our formula is exact for any tetration base which is not coprime to 10, although a maximum gap equal to V(a) + 1 digits (where V(a) indicates the congruence speed of a) can occur, in the worst-case scenario, between the given upper and lower bound.
Category: Number Theory
[25] viXra:2112.0161 [pdf] replaced on 2022-01-06 07:34:29
Authors: Theophilus Agama
Comments: 5 Pages. The results have now been made unconditional based on referee report
In this paper we introduce and develop the method of diagonalization of functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems requiring to determine if the equations of the form $f(n)+k=m^2$ has a finite number of solutions $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$.
Category: Number Theory
[24] viXra:2112.0159 [pdf] submitted on 2021-12-30 14:20:05
Authors: Bertrand Wong
Comments: 8 Pages.
This paper brings up a few possible approaches to solving the twin primes conjecture. [Published in international mathematics journal. Acknowledgments: The author is thankful to the referees and to the Editor-in-Chief for their insightful comments which led to an improvement in this paper.]
Category: Number Theory
[23] viXra:2112.0153 [pdf] submitted on 2021-12-29 21:22:08
Authors: Waldemar Puszkarz
Comments: 8 pages, extends work originally posted on viXra: https://vixra.org/abs/1804.0416.
Computer experiments show that singly even multiples of 6 sur-rounded by prime pairs exhibit a larger ratio of nonsquarefree to squarefree multiples than generic singly even multiples of 6, a bias of ca 10.6% measured against the expected value. The same bias occurs for isolated primes next to singly even multiples of 6; here the deviation from the expected value is ca 3.3% of this value. The expected value of the ratio of singly even to doubly even nonsquarefree multiples of 6 also differs from values found experimentally for prime pairs centered on such multiples or isolated primes next to them. For pairs, this ratio exceeds its unbiased value by ca 6.2%, for isolated primes by ca 2.0%. The values cited are for the first 10^10 primes, the largest range we investigated. This paper broadens our recent study of a newly found bias in the distribution of primes by examining singly and doubly even multiples of 6. In particular, it shows that for primes centered on or next to singly even multiples of 6, the statistical biases in question are more pronounced than in the general case studied by us before.
Category: Number Theory
[22] viXra:2112.0145 [pdf] submitted on 2021-12-28 20:39:29
Authors: Aric B. Canaanie
Comments: 10 Pages.
The central idea of this article is to introduce and prove a special form of the zeta function as proof of Riemann’s last theorem. The newly proposed zeta function contains two sub functions, namely f1(b,s) and f2(b,s) . The unique property of zeta(s)=f1(b,s)-f2(b,s) is that as tends toward infinity the equality zeta(s)=zeta(1-s) is transformed into an exponential expression for the zeros of the zeta function. At the limiting point, we simply deduce that the exponential equality is satisfied if and only if real(s)=1/2 . Consequently, we conclude that the zeta function cannot be zero if real(s)=1/2 , hence proving Riemann’s last theorem.
Category: Number Theory
[21] viXra:2112.0142 [pdf] replaced on 2021-12-31 21:23:54
Authors: Miroslav Sukenik, Magdalena Sukenikova
Comments: 4 Pages. final version
The article examines the control function in relation to the distribution of Zeros on the critical line x = 0,5. To confirm its importance, it will be necessary to perform a large number of statistical analyzes of the distribution of non-trivial zero points of the Riemann Zeta function.
Category: Number Theory
[20] viXra:2112.0140 [pdf] submitted on 2021-12-27 13:44:39
Authors: Marko V. Jankovic
Comments: 7 Pages.
In this paper, it is going to be explained how all natural numbers can be presented as a multiplication tensor or Mtensor. This comes as an extension of the number line (that is graphical presentation of the numbers that is suitable for
graphical presentation of addition/subtraction) and facilitates reasoning related to sieves, multiplication, prime and
composite numbers and so on. Here, the new representation will be used to calculate number of rational numbers.
Category: Number Theory
[19] viXra:2112.0137 [pdf] submitted on 2021-12-26 09:29:35
Authors: Mar Detic
Comments: 1 Page.
By using the Infinite/Harmonic series(partial) we prove if n is prime and show factors if n is not prime at the same time ; without trial division or modulo.
Category: Number Theory
[18] viXra:2112.0103 [pdf] submitted on 2021-12-19 08:02:17
Authors: Denis Gallet
Comments: 4 Pages.
In this paper, I study particulary logarithmics integrals.
Category: Number Theory
[17] viXra:2112.0093 [pdf] replaced on 2021-12-21 08:42:17
Authors: Bassam Abdul-Baki
Comments: 5 Pages.
An interesting if not impractical way of primality testing and factoring a number using Pascal’s Triangle.
Category: Number Theory
[16] viXra:2112.0081 [pdf] submitted on 2021-12-15 22:52:13
Authors: Masami Yamane, Kenji Matsuura, Seiichi Koshiba
Comments: 4 Pages.
In this note, we are interested in the last numbers of positive integers; for example, for 20211206, the last number is 6, typically we note that for any
positive integer a, the last numbers of a5 and a are the same.
Category: Number Theory
[15] viXra:2112.0074 [pdf] submitted on 2021-12-14 18:22:12
Authors: V. Barbera
Comments: 3 Pages.
This paper presents the use of a specific wheel factorization sieve algorithm in some applications.
Category: Number Theory
[14] viXra:2112.0070 [pdf] replaced on 2024-10-16 13:54:09
Authors: Andrea Berdondini
Comments: 6 Pages.
Any result can be generated randomly and any random result is useless. Traditional methods define uncertainty as a measure of the dispersion around the true value and are based on the hypothesis that any divergence from uniformity is the result of a deterministic event. The problem with this approach is that even non-uniform distributions can be generated randomly and the probability of this event rises as the number of hypotheses tested increases. Consequently, there is a risk of considering a random and therefore nonrepeatable hypothesis as deterministic. Indeed, it is believed that this way of acting is the cause of the high number of non-reproducible results. Therefore, we believe that the probability of obtaining an equal or better result randomly is the true uncertainty of the statistical data. Because it represents the probability that the data is useful and therefore the validity of any other analysis depends on this parameter.
Category: Number Theory
[13] viXra:2112.0067 [pdf] replaced on 2022-12-31 22:06:07
Authors: Abdelmajid Ben Hadj Salem
Comments: 70 Pages. A new chapter is added. Comments welcome.
This report presents a collection of some tentatives to obtain a final proof of the Riemann Hypothesis. The last paper of the report is submitted to a mathematical journal for review.
Category: Number Theory
[12] viXra:2112.0061 [pdf] submitted on 2021-12-12 09:21:37
Authors: Theophilus Agama
Comments: 5 Pages.
In this paper we introduce a multivariate version of circles of partition introduced and studied in \cite{CoP}. As an application we prove a weaker general version of the Erd\H{o}s-Tur\'{a}n additive base conjecture. The actual Erd\H{o}s-Tur\'{a}n additive base conjecture follows from this general version as a consequence.
Category: Number Theory
[11] viXra:2112.0056 [pdf] submitted on 2021-12-11 20:11:31
Authors: Kurmet Sultan
Comments: 2 Pages.
The article provides a proof of the ABC conjecture, obtained using the generalized Euler and Fermat theorems, as well as modular arithmetic.
Category: Number Theory
[10] viXra:2112.0050 [pdf] submitted on 2021-12-10 21:46:31
Authors: V. Barbera
Comments: 2 Pages.
This paper presents an exact elementary formula for the prime-counting function and an exact formula for
counting pairs of twin prime numbers.
Category: Number Theory
[9] viXra:2112.0044 [pdf] submitted on 2021-12-09 20:56:17
Authors: Peter G. Bass
Comments: 4 Pages.
The Legendre Conjecture is herein proved by analysing the difference between the Prime Number Theorem for adjacent squares, and also by estimating the number ofncomposites between adjacent squares using a slight variation of the Prime Number
Theorem.
Category: Number Theory
[8] viXra:2112.0036 [pdf] submitted on 2021-12-07 21:13:22
Authors: Mohamed Azzedine
Comments: 8 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
This is a Direct proof of Fermat's Last Theorem based on Even/Odd parity of numbers. It is short,direct and comprehensible by student in Mathematics and lovers of Mathematics.
Category: Number Theory
[7] viXra:2112.0035 [pdf] submitted on 2021-12-07 21:17:33
Authors: Mohamed Azzedine
Comments: 6 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
Direct proof of fermat's Last Theorem (x^n +y^n =z^n)based on Induction on z not on n. it is short, direct and comprehensible by student in Mathematics and lovers of Mathematics. It use mathematical tools of Fermat's era.
Category: Number Theory
[6] viXra:2112.0033 [pdf] replaced on 2021-12-15 04:45:36
Authors: Shazly Abdullah
Comments: 9 Pages.
.
In this study we used an algebraic method that uses elementary algebra and binomial theorem. To create series We used these series to study the prime numbers of the from ,We found several characteristics . for example, we proved If , p prime number and where then ( ) .We also obtained several results in finite series
Category: Number Theory
[5] viXra:2112.0027 [pdf] submitted on 2021-12-06 09:33:14
Authors: Theophilus Agama
Comments: 4 Pages.
In this paper we prove the binary Goldbach conjecture. By exploiting the language of circles of partition, we show that for all sufficiently large $n\in 2\mathbb{N}$
\begin{align}
\# \left \{p+q=n|~p,q\in \mathbb{P}\right \}>0.\nonumber
\end{align}This proves that every sufficiently large even number can be written as the sum of two prime numbers.
Category: Number Theory
[4] viXra:2112.0022 [pdf] submitted on 2021-12-05 19:55:49
Authors: Jose Risomar Sousa
Comments: 6 Pages.
A new relation between the Lerch's transcendent, $\Phi$, and the Hurwitz zeta, $\zeta(k,b)$, at the positive integers is introduced. It is derived simply by inverting the relation presented in the precursor paper with one of two approaches (its generating function or the binomial theorem). This enables one to go from Lerch as a function of Hurwitz zetas (of different orders), to Hurwitz as a function of Lerches. A special case of this new functional equation is a relation between the Riemann's zeta function and the polylogarithm.
Category: Number Theory
[3] viXra:2112.0021 [pdf] submitted on 2021-12-04 19:21:46
Authors: Shazly Abdullah
Comments: 8 Pages. [Corrections made by viXra Admin]
In this work we used an algebraic method that uses elementary algebra . To create series. We used the series and Euler function φ(n) to find solutions to some types of Diophantine equations such as p = dn - n + 1. We found a relationship between the solutions of the Diophantine equations and solutions of some types of congruences that use the φ(n) function. This relationship is the results that relate the solutions of congruence to the solution of the equations
Category: Number Theory
[2] viXra:2112.0014 [pdf] submitted on 2021-12-03 15:11:06
Authors: Shazly Abdullah
Comments: 9 Pages. 9
In this study we used an algebraic method that uses elementary algebra and binomial theorem. To create series We used these series to study the prime numbers of the form , We found several characteristics . for example, we proved If , p prime number and where then ( ) .We also obtained several results in finite series.
Category: Number Theory
[1] viXra:2112.0004 [pdf] submitted on 2021-12-01 20:42:53
Authors: Giovanni Di Savino
Comments: 2 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
With the Collatz algorithm it is not possible to process all natural numbers because we do not know: quantities and values of even and odd numbers and all their factors. From Tartaglia's triangle we can detect odd numbers which are the sum of the results of the infinite powers of 2 which have an even index and which are also equal to the previous odd * 4 + 1. These are all the odd numbers that * 3 + 1 generate an even number that is the result of a base power 2 and even index 2 ^ (2 * n≥1) and that, the nth half, ends at 1 because ½ of 2 ^ 1 = 2 ^ 0 = 1.
Category: Number Theory
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