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[21] viXra:2408.0107 [pdf] submitted on 2024-08-25 11:17:20
Authors: Timothy Jones
Comments: 7 Pages.
We define a Goldbach table as a table consisting of two rows. The lower row counts from 0 to any n and and the top row counts down from 2n to n. All columns will have all numbers that add to 2n. Using a sieve, all composites are crossed out and only columns with primes are left. We then define a novel prime decimal system: it gives for every n remainders when n is divided by all primes less than the n. This suggests linear functions, the divisions used can give another perspective on all the column pairs. The inverses of these functions when put into tabular form give symmetries that suggest Goldbach's conjecture is correct.
Category: Number Theory
[20] viXra:2408.0106 [pdf] submitted on 2024-08-25 12:43:39
Authors: Andrey V. Voron
Comments: 4 Pages.
Based on the theoretical analysis of 4×4 pandiagonal squares, their "structure" features are shown: the invariants of the structure of 4×4 pandiagonal squares are pairs of numbers equal in sum to one of the two Fibonacci numbers — 13 or 21. It is revealed that any variant of the set of six digits of the Durer square and similar 4×4 pandiagonal squares, forming a continuous symmetric configuration, is equal in total to the integer 51. A geometric figure "cube in a cube" is constructed, which has the properties of the "golden symmetry" of 4×4 pandiagonal squares. All the numbers of the diagonals of the cube have the properties of "golden symmetry" (two numbers form in one case the total number 13, in the other — 21), and all planes having 4 angles (numbers) both the inner and outer squares of the geometric figure form a total of the Fibonacci number — 34.
Category: Number Theory
[19] viXra:2408.0092 [pdf] submitted on 2024-08-22 19:50:32
Authors: Pedro Caceres
Comments: 45 Pages.
Prime numbers are the atoms of mathematics and mathematics is needed to make sense of the real world. Finding the Prime number structure and eventually being able to crack their code is the ultimate goal in what is called Number Theory. From the evolution of species to cryptography, Nature finds help in Prime numbers. One of the most important advances in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" (On the number of primes less than a given quantity).In that paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function defined by:[RZF] ζ(z)=∑(n=1,∞) 1/n^z Where ζ(z) is a function of a complex variable z that analytically continues the Dirichlet series. Riemann also formulated a conjecture about the location of the zeros of RZF, which fall into two classes: the "trivial zeros" -2, -4, -6, etc., and those whose real part lies between 0 and 1. Riemann's conjecture Riemann hypothesis [RH] was formulated as this:[RH]The real part of every nontrivial zero z* of the RZF is 1/2.Proving the RH is, as of today, one of the most important problems in mathematics. In this paper we will provide proof of the RH. The proof of the RH will be built following these five parts:PART 1:Description of the Riemann Zeta Function RZF ζ(z) - Introducing s limit and an approximationPART 2: The C-transformation. An artifact to decompose ζ(z) PART 3: Application of the C-transformation to f(z)=1/x^z in Re(z)≥0 to obtain ζ(z)=X(z)-Y(z) - Decomposition of ζ(z)=X(z)-Y(z) - Analysis of X(z),|X(z)|,|X(z)|^2 - Analysis of Y(z),|Y(z)|,|Y(z)|^2PART 4: Proof of the Riemann Hypothesis - Analysis of the values of z such that X(z)=Y(z), that equates to ζ(z)=0 - Proof that |X(z)|=|Y(z)| only if Re(z)=1/2 - Conclude that ζ(z)=0 only if Re(z)=1/2 for Re(z)≥0PART 5: On the distribution of the non-trivial zeros of Zeta in the critical line α= 1/2. - Algorithm N1, Algorithm H1, Algorithm H2
Category: Number Theory
[18] viXra:2408.0089 [pdf] submitted on 2024-08-20 00:54:20
Authors: Seiji Tomita
Comments: 3 Pages.
In this paper, we will construct a new family of elliptic curves with the rank of at least three arising from quartic curves.
Category: Number Theory
[17] viXra:2408.0080 [pdf] submitted on 2024-08-18 22:05:02
Authors: Zhiyang Zhang
Comments: 15 Pages.
The counterexample of the Riemann hypothesis causes a significant change in the image of the Riemann Zeta function, which can be distinguished using mathematical judgment equations. The counterexamples can be found through this equation.
Category: Number Theory
[16] viXra:2408.0079 [pdf] submitted on 2024-08-18 22:00:43
Authors: Khazri Bouzidi Fethi
Comments: 4 Pages. (Note by viXra Admin: Article title should be above the author's name)
We will use another relationship between the estimation of gauss for the prime numbers and the zeta function to arrive at affirming the reimann hypothesis.
Category: Number Theory
[15] viXra:2408.0071 [pdf] replaced on 2025-07-28 08:45:43
Authors: V. Barbera
Comments: 6 Pages.
This paper presents an extension of the left-to-right binary method to perform modularexponentiation b^e (mod m) by representing the exponent in base 2^d.
Category: Number Theory
[14] viXra:2408.0059 [pdf] submitted on 2024-08-15 09:50:08
Authors: Bryce Petofi Towne
Comments: 31 Pages.
The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line ( Re(s) = frac{1}{2} ) in the complex plane. This paper explores an alternative geometric approach by analyzing the zeta function and its non-trivial zeros in polar coordinates. Transforming the problem into this framework reveals a natural symmetry about the polar axis, which corresponds to the critical line in Cartesian coordinates.We demonstrate that the (Xi(s)) function, a redefined version of the zeta function, retains the symmetry ( Xi(s) = Xi(1 - s) ) in polar coordinates, supporting the hypothesis that non-trivial zeros must lie on the critical line.This geometric perspective suggests a potential simplification in verifying the Riemann Hypothesis and offers new insights into the distribution of non-trivial zeros.
Category: Number Theory
[13] viXra:2408.0056 [pdf] replaced on 2025-11-07 04:13:21
Authors: WenBin Hu
Comments: 6 Pages.
Goldbach's conjecture has been around for more than 300 years and the twin prime conjecture for more than 160 years. Both remain unsolved. Both conjectures are important number theory conjectures for studying prime numbers. This article proposes a method of sequence shift to solve. This may be a good method and seems quite easy to understand.
Category: Number Theory
[12] viXra:2408.0055 [pdf] submitted on 2024-08-14 07:40:49
Authors: Daniil Krizhanovskyi
Comments: 9 Pages.
The advent of quantum computing represents a paradigm shift with profound implications for the field of cryptography. Quantum algorithms, particularly Shor's algorithm, threaten to undermine the security foundations of traditional cryptographic schemes such as RSA, ECC, and DSA, which rely on the computational difficulty of integer factorization and discrete logarithms. As these algorithms become obsolete in the face of quantum capabilities, there is an urgent need for cryptographic systems that can withstand quantum-based attacks. In response to this looming threat, this paper introduces the Quantum Cryptographic Toolkit (QCT), a robust and versatile framework designed to facilitate the development, testing, and deployment of quantum-resistant cryptographic algorithms. The QCT integrates a diverse set of post-quantum cryptographic algorithms, including lattice-based methods like NewHope, code-based approaches exemplified by the McEliece cryptosystem, and isogeny-based cryptography, such as SIKE. Each of these algorithms is implemented with a focus on maintaining security even in the face of quantum computing advancements, addressing both theoretical and practical challenges. The toolkit is structured to be modular and extensible, allowing researchers and developers to seamlessly incorporate additional algorithms and cryptographic primitives as the field evolves. This paper details the design principles underlying the QCT, emphasizing the importance of modularity, extensibility, and performance optimization. We discuss the implementation strategies employed to ensure the toolkit's effectiveness across a range of cryptographic scenarios, from key exchange protocols to encryption and digital signatures. A comprehensive security analysis is provided, highlighting the resistance of each algorithm to quantum attacks, and comparing their performance to other post-quantum cryptographic solutions. In addition to the security analysis, we include extensive performance benchmarks that evaluate the computational efficiency, memory usage, and scalability of the algorithms within the QCT. These benchmarks demonstrate the practical viability of the toolkit for real-world applications, offering insights into the trade-offs between security and performance that are inherent in post-quantum cryptography. The results indicate that the QCT not only meets the stringent security requirements
Category: Number Theory
[11] viXra:2408.0051 [pdf] submitted on 2024-08-13 02:38:36
Authors: Nikhil Datar
Comments: 10 Pages. (Note by viXra Admin: Please cite and list scientific references)
This submission gives a proof of existence of infinite twin primes. The basic principle used here is of Fundamental Principle of Counting and Excel has been as a major tool for explanation
Category: Number Theory
[10] viXra:2408.0045 [pdf] submitted on 2024-08-11 21:06:58
Authors: Adrian M. Stokes
Comments: 5 Pages.
The prime numbers ≥ 5 within a finite sequence of natural numbers can be found by calculating all of the values given by 6n±1 that fall within the sequence and subtracting the composites given by (6n_1±1)(6n_2±1), where n is a natural number. A test model for finding primes based on this method uses three reference sub-set multiplication tables to calculate composites and then matches these to the corresponding values in the sets {6n-1} and {6n+1}. The unmatched numbers are primes. Although this model provides a useful proof of concept, it is impractical at scale. A new method that replaces the sub-set tables with an equation to calculate 6n±1 composite pairs forms the basis of an improved model using sieve methodology.
Category: Number Theory
[9] viXra:2408.0041 [pdf] submitted on 2024-08-09 08:47:16
Authors: Bryce Petofi Towne
Comments: 14 Pages.
This paper investigates the symmetry of non-trivial zeros of the Riemann zeta function (zeta(s)) through geometric analysis in polar coordinates. By transforming the complex number (s = sigma + it) into polar form, we demonstrate that the symmetry about the critical line (sigma = frac{1}{2}) necessitates (sigma = frac{1}{2}) for all non-trivial zeros. Numerical simulations further confirm the accuracy and consistency of this geometric approach. And we introduce a formula for the distribution pattern of all non-trivial zeros:[zetaleft(sqrt{frac{1}{4} + t^2} , e^{i arctan(2t)}ight) = 0]where: [r = sqrt{frac{1}{4} + t^2} quad text{and} quad theta = arctan(2t)]
Category: Number Theory
[8] viXra:2408.0028 [pdf] submitted on 2024-08-08 18:54:41
Authors: Timothy Jones
Comments: 4 Pages.
We use a series of tables with an induction argument to show Goldbach's conjecture is true.
Category: Number Theory
[7] viXra:2408.0025 [pdf] submitted on 2024-08-07 18:45:14
Authors: Edgar Valdebenito
Comments: 5 Pages.
We give some formulas related to the Fransén-Robinson constant F=2.80777024...
Category: Number Theory
[6] viXra:2408.0024 [pdf] submitted on 2024-08-06 20:41:28
Authors: Dmitri Martila, Stefan Groote
Comments: 3 Pages.
A criterion given by Jean-Louis Nicolas is used to offer a proof for the Riemann Hypothesis in a straightforward way.
Category: Number Theory
[5] viXra:2408.0021 [pdf] submitted on 2024-08-06 20:32:20
Authors: Khazri Bouzidi Fethi
Comments: 3 Pages.
��(��)The TNP prime number counting function��(��)Gaussian approximation for prime numbersDensity sum error for x superior 600 So 0 ≤ �� — �� ≤ 0.04.
Category: Number Theory
[4] viXra:2408.0019 [pdf] submitted on 2024-08-05 07:13:49
Authors: Bryce Petofi Towne
Comments: 16 Pages.
This paper investigates the non-trivial zeros of the Riemann zeta function using polar coordinates. By transforming the complex plane into a polar coordinate system, we provide a geometric perspective on the distribution of non-trivial zeros. We focus on the key formula:[zetaleft(sqrt{frac{1}{4} + t^2} , e^{i arctan(2t)}ight) = 0]This formula reveals the distribution pattern of non-trivial zeros and supports the hypothesis that (sigma) must be (frac{1}{2}) and is a constant.
Category: Number Theory
[3] viXra:2408.0018 [pdf] submitted on 2024-08-05 21:05:43
Authors: Oliver Couto
Comments: 5 Pages.
Historicaly, we note that finding a prametrization of degree six has not been easy. In the below paper the author has followed in the footsteps of below mentioned paper, ref. no. (1). In ref. no.(1), Ajai Choudhry on page 356 has parametrized equation: m(abc)(ab+bc+ca)(a+b+c)=n(pqr)(pq+pr+qr)(p+q+r), where, (m,n)=(3,32) using one parameter. In the below paper the author has parametrized the said equation for (m,n)=(1,1), but with two parameters. The trick used is to split equation (1) into three parts, so as to balance the coefficents (m,n) & such that, (m,n)=[(uvw),(xyz)]. And [(uvw),(xyz)] =(72,72), & hence they cancel each other out.
Category: Number Theory
[2] viXra:2408.0007 [pdf] submitted on 2024-08-02 20:06:01
Authors: Rédoane Daoudi
Comments: 7 Pages.
Here I present several formulas and conjectures on prime numbers. I’m interested in studying prime numbers, Euler’s totient function and sum of the divisors of natural numbers.
Category: Number Theory
[1] viXra:2408.0003 [pdf] submitted on 2024-08-01 12:08:51
Authors: Timothy Jones
Comments: 4 Pages.
We introduce a Goldbach table. It consists of two rows. A bottom row counts from zero to a given n and the top counts from the right from n to 2n. The columns generated give all the whole numbers that add to 2n. We confirm that using a sieve, we do always seem to get top and bottom primes that show Goldbach's conjecture is true for the particular 2n depicted by this table. Next we cumulatively depict these tables and we see some interesting patterns. We can infer that all prime pairs will occur in one of these tables. We also see diagonal prime lines that seem to start and stop in symmetrical ways. These patterns suggest that for any even number we can choose a column and then find a prime pair.
Category: Number Theory
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