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[19] viXra:2406.0190 [pdf] submitted on 2024-06-30 22:18:16
Authors: Taha Sochi
Comments: 178 Pages.
This is the second volume of my book "Notes and Problems in Number Theory".
Category: Number Theory
[18] viXra:2406.0187 [pdf] submitted on 2024-06-30 16:49:47
Authors: Taha Sochi
Comments: 237 Pages.
This book is the first volume of a collection of notes and solved problems about number theory. Like my previous books, maximum clarity was one of the main objectives and criteria in determining the style of writing, presenting and structuring the book as well as selecting its contents.
Category: Number Theory
[17] viXra:2406.0173 [pdf] submitted on 2024-06-28 20:55:24
Authors: Athon Zeno, Aeon Zeno
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific references)
This paper presents a new perspective on prime number distribution, proposing a fractal-like structure that manifests at multiple scales. We introduce a mathematical framework, utilizing modular arithmetic and the Chinese Remainder Theorem, to prove self-similarity in prime distribution. Our model offers potential insights into the Riemann Hypothesis and suggests new approaches to understanding prime number gaps. Computational evidence up to 10^9 demonstrates consistent fractal dimensions across scales, agreement with predicted scaling factors, and self-similar prime gap distributions, strongly supporting our theoretical framework.
Category: Number Theory
[16] viXra:2406.0172 [pdf] submitted on 2024-06-28 20:53:57
Authors: Aziz Arbai, Amina Bellekbir
Comments: 8 Pages.
We research and explicitly expose example of an infinity of zeros (C(r+ic)=0) of RH (The Riemann hypothesis) in the critical line (having for real part r= 1/2 and c=[+or-pi/4+2kpi]/ln(2)). So there is infinity of no- trivial zeros of Riemann’s zeta function which have the real part equal to 1/2, which shows (using simple mathematics baggage) Hardy and Littlewood Theorem and give as a hope that the Riemann’s Conjecture would be true....
Category: Number Theory
[15] viXra:2406.0154 [pdf] submitted on 2024-06-25 01:09:29
Authors: Seiji Tomita
Comments: 2 Pages.
In this paper, we prove that the positive integer solutions of the equation x^2 +7 = 2^n are x = 1, 3, 5, 11, 181, corresponding to n = 3, 4, 5, 7, 15.
Category: Number Theory
[14] viXra:2406.0150 [pdf] submitted on 2024-06-25 13:30:58
Authors: Dmitri Martila
Comments: 2 Pages.
Suppose the Riemann Zeta function is multiplied by two arbitrary functions, and the resulting functions' values are equated at symmetrical points concerning the critical line Re s = 1/2. In that case, the resulting system of fourequations has to give the positions of the Zeta function's zeros. However, since the functions are arbitrary, the positions of the zero places are arbitrary, making a zero coincide with non-zero. Hence, the Riemann Hypothesis that the only zeroes are those on the critical line is true. This simple text is proof of the Riemann hypothesis, Generalized Riemann hypothesis and Extended Riemann hypothesis with accordingfunctions.
Category: Number Theory
[13] viXra:2406.0134 [pdf] replaced on 2025-07-26 07:34:33
Authors: Marcin Barylski
Comments: 4 Pages. Updating references, fixing two typos in the main text.
One of the most famous unsolved problems in mathematics is Collatz conjecture which is claiming that all positve numbers subjected to simple 3x + 1/2 formula will eventually result in 1, with only one known cycle (1, 4, 2, 1) present in the calculations. This work is devoted to finding cycles in other interesting sequences of integer numbers, constructed with the use of some aspect of primality test.
Category: Number Theory
[12] viXra:2406.0105 [pdf] replaced on 2024-07-05 16:23:37
Authors: Timothy Jones
Comments: 3 Pages. Got some suggestions for improving.
We clarify and strengthen Hardy's footnote proof of an essential step in his proof of the transcendence of pi. We show that ri is algebraic if and only if r is algebraic.
Category: Number Theory
[11] viXra:2406.0077 [pdf] replaced on 2025-03-16 04:08:29
Authors: Hajime Mashima
Comments: 29 Pages.
Modulo not divisible by xyz and possible expansions.
Category: Number Theory
[10] viXra:2406.0073 [pdf] submitted on 2024-06-14 04:36:05
Authors: Seiji Tomita
Comments: 5 Pages.
In this paper, we proved that there are infinitely many integer solutions of X^6 + Y^6 = W^n + Z^n, n=2,3,4.
Category: Number Theory
[9] viXra:2406.0072 [pdf] replaced on 2024-12-09 04:41:28
Authors: Junho Eom
Comments: 13 Pages. 1 figure
The core of this paper is to reveal the structural necessity that causes primes and new primes to form a symmetry, occurring from the cause-and-effect relationship between primes and composites. Regarding the boundary, if an arbitrary integer n is chosen, the set of consecutive numbers from 0 to n is defined as the 1st boundary and it extends using an arithmetic sequence with n elements but limits to n^2. After selecting n, therefore, n boundaries are generated from the 1st to nth, and each boundary contains n elements. Each prime wave in the 1st boundary connects to the composites that use the prime as a factor, and the remaining numbers between the 2nd and nth boundaries on the x-axis are all new primes in Series I. Under this condition, the primes in the 1st boundary and the new primes in the 2nd boundary form symmetry around the midpoint of even 2n caused by the asymmetry between primes and composites, Goldbach’s conjecture is satisfied in Series II and III. Therefore, Series IV explains the necessity for the primes and new primes to form a structural symmetry using 2 and 3, and discusses how this symmetry repeats at intervals of 30, generated by 5.
Category: Number Theory
[8] viXra:2406.0070 [pdf] replaced on 2024-07-19 05:10:24
Authors: Bryce Petofi Towne
Comments: 22 Pages.
This paper presents an approach to analyzing the non-trivial zeros of the Riemann zeta function using polar coordinates. We investigate whether the real part of all non-trivial zeros can be determined to be a constant value. By transforming the traditional complex plane into a polar coordinate system, we recalculated and examined several known non-trivial zeros of the zeta function. Our findings provide an alternative framework for understanding this profound mathematical conjecture.Through mathematical proof and leveraging analytic continuation and holomorphic function theory, we explore the nature of (sigma) in the polar coordinate system. This analysis transforms the problem into a geometric one, allowing for simpler and more intuitive calculations. This approach provides a step towards an alternative understanding of the properties of the Riemann zeta function's non-trivial zeros. The findings of this work indicates that wit this geometric perspective, the Riemann Hypothesis holds true.
Category: Number Theory
[7] viXra:2406.0054 [pdf] submitted on 2024-06-12 05:39:22
Authors: Seiji Tomita
Comments: 3 Pages.
In this paper, we prove that there are infinitely many integers that can be expressed as the sum of four cubes of polynomials.
Category: Number Theory
[6] viXra:2406.0045 [pdf] submitted on 2024-06-10 16:52:07
Authors: Taha Sochi
Comments: 75 Pages.
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in particular to young "mathematicians" dealing mostly with Diophantine equations at elementary level of number theory (noting that familiarity with elementary number theory is generally required).
Category: Number Theory
[5] viXra:2406.0030 [pdf] replaced on 2024-06-14 21:25:31
Authors: Bassera Hamid
Comments: 1 Page. Sent to American Mathematical Society in June 05 2024
In this article I try to make my modest contribution to the proof of Goldbach’s conjecture and I propose to simply go through its negation.
Category: Number Theory
[4] viXra:2406.0025 [pdf] submitted on 2024-06-06 05:28:47
Authors: Seiji Tomita
Comments: Pages.
In this paper, we proved that there are infinitely many integers n such that a+b+c=1/a+1/b+1/c=n has infinitely many rational solutions.
Category: Number Theory
[3] viXra:2406.0020 [pdf] submitted on 2024-06-05 19:56:21
Authors: Budee U. Zaman
Comments: 9 Pages.
This paper presents a new proof of the Goldbach conjecture, which is a well-known problem originating from number theory that was proposedby Christian Goldbach back in 1742. Our way gives a simple but deep understanding of the even integers can be written as the sum of two prime numbers Through examining fully we show that every other even integer larger than two will essentially represent itself in form adding up two prime numbers. The revelation of a straightforward and elegant line to this enduring conjecture comes from the use of basic number theory concepts such as; by going a step further and coming up with creative strategies. There is more evidence and sound payments she makes for her assertion as we continue.The centuries-old mathematical puzzle has been solved paving way for the exploration of new possibilities in number theory and we are grateful for the perspective and the persistence accorded us by God, which enabled us to reach this milestone.
Category: Number Theory
[2] viXra:2406.0016 [pdf] submitted on 2024-06-04 13:30:29
Authors: Ricardo Gil
Comments: 3 Pages.
The distribution and density of these zeros affect the error term in the Prime Number Theorem. If the Riemann Hypothesis holds, it implies a tighter error bound in the Prime Number Theorem.
Category: Number Theory
[1] viXra:2406.0008 [pdf] submitted on 2024-06-02 22:28:31
Authors:
Comments: 4 Pages. (Author name added to the article by viXra Admin as required)
Since Fermat’s equation,[(a^3+b^3 )=(c)^3 ]does not have a solution,we are considering the below two Diophantine equations:∶(a^3+b^3 )=w(c)^3 -----(1)(a^3-b^3 )=w(c)^3 -----(2)Also, equation (2) above has been discussed in the book by Tito Piezas (Ref. # 3).
Category: Number Theory
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