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| viXra.org > Number Theory > 2007 |
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[22] viXra:2007.0243 [pdf] submitted on 2020-07-31 20:07:02
Authors: Aziz Arbai
Comments: 6 Pages.
We propose the demonstration of the Riemann hypothesis, thus we explicitly expose (an infinity of complex numbers α=(1/2)+ic ) the zeros (having for real part 1/2) of the Zeta function of Riemann, which will also give exactly the distribution and location of the prime numbers.
Category: Number Theory
[21] viXra:2007.0220 [pdf] replaced on 2020-08-15 04:22:51
Authors: A. A. Frempong
Comments: 16 Pages. Copyright © by A. A. Frempong
After proving the strong Goldbach conjecture (viXra:2006.0226), the author, in this paper, covers both the weak Goldbach conjecture and the strong Goldbach conjecture. The strong Goldbach conjecture states that every even integer greater than 4 can be expressed as the sum of two odd primes. The weak Goldbach conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. The approach in the coverage of the weak Goldbach conjecture is similar to the approach used in proving the strong conjecture. However, two approaches for producing Goldbach partitions for the weak conjecture are covered. In the first approach, one applies the principles used in finding partitions for the strong conjecture. Beginning with the partition equation, 9 = 3 + 3 + 3, and applying the addition of a 2 to both sides of this equation, and subsequent equations, one obtained Goldbach partitions for 26 consecutive odd integers. In the second approach, one produces the partitions from the partitions of the strong conjecture by adding a 3 to both sides of a strong conjecture partition equation. For the strong conjecture, one will begin with the partition equation 6 = 3 + 3, and apply the addition of 2 to both sides of the equation to produce the partition for the next even number, 8. From the partition equation, 8 = 5 + 3, one will repeat the 2-addition process to obtain the partition for the next even integer, 10. From the partition for 10, the process can continue indefinitely. This repetitive process was compared to the repetitive process in compound interest calculations. It is shown that given an equation for a Goldbach partition, one can produce a Goldbach partition for any even integer greater than 4 as well as produce a partition for any odd integer greater than 7. A consequent generalized procedure also produced Goldbach partitions for non-consecutive even and non-consecutive odd integers. In addition to directly producing partitions of the strong conjecture, one can also produce partitions of the strong conjecture from the partitions of the weak conjecture and vice versa. Formulas derived for the Goldbach partitions show that every even integer greater than 4 can be written as the sum of two odd prime integers; and also that every odd integer greater than 7 can be written as the sum of three odd prime integers. Importantly, in addition to showing that the Goldbach conjectures are true, this paper shows how to produce Goldbach partitions.
Category: Number Theory
[20] viXra:2007.0205 [pdf] replaced on 2020-08-21 09:45:38
Authors: Yukihiro Sano
Comments: 31 Pages.
In the Ulam spiral, there are places where prime numbers appear continuously on line. Integers are arranged in a square spiral in the Ulam spiral. I thought that if integers are arranged differently, other continuous prime numbers would appear. Therefore, I arrange integers in the angles of 45, 90, 135, 180, 225, 270, 315, 153, 160 degrees, etc.. Then, prime numbers appeared continuously on line. And usually, integers are arranged, but I wonder what would happen if I arranged odd numbers. I arrange odd numbers in the angles of 45, 90, 135, 180, 225, 270, 315, 360, 153, 160 degrees, etc.. Then, twin prime numbers appeared continuously on line etc.. I found many polynomials generating 14 to 4 consecutive twin prime numbers.
Category: Number Theory
[19] viXra:2007.0196 [pdf] submitted on 2020-07-24 19:32:55
Authors: Mohammed zohal
Comments: 18 Pages. in this papers there is a proof of the Twin Prime conjecture
We will prove the next results :
1. there exist infinite twin primes .
2. there exist infinite cousin primes .
3. The cousin primes are equivalent to twin primes in infinity.
Category: Number Theory
[18] viXra:2007.0141 [pdf] submitted on 2020-07-17 21:28:29
Authors: Miguel Cerdá Bennassar
Comments: Pages.
No se puede entender este escrito si no se conocen los anteriores de Agosto y Noviembre 2019. Para visualizar los gráficos, aconsejo descargar y ampliar el pdf. En la siguiente tabla están los números pares y los resultados posibles de dividirlos entre 2. Los de color verde solamente admiten una división, mientras que los de color rojo admiten más de una. El color amarillo señala los números pares que son el resultado de aplicar 3n+1 a los números impares n.
(It is not possible to understand this writing if the previous ones of August and November 2019 are not known. To view the graphics, I recommend downloading and enlarging the pdf. In the following table are the even numbers and the possible results of dividing them by 2. Those of green color only admit one division, while those of red color admit more than one. The yellow color indicates the even numbers that are the result of applying 3n + 1 to the odd numbers n.)
Category: Number Theory
[17] viXra:2007.0128 [pdf] submitted on 2020-07-16 19:49:36
Authors: Theophilus Agama
Comments: 6 Pages.
Under the assumption that $\sum \limits_{n\leq N}\Upsilon(n)\Upsilon(N-n)>0$, we show that for all even number $N>6$ \begin{align} \sum \limits_{n\leq N}\Upsilon(n)\Upsilon(N-n)=(1+o(1))K\sum \limits_{p|N}\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)\nonumber
\end{align}for some constant $K>0$, and where $\Upsilon$ and $\Lambda_{0}$ denotes the master and the truncated Von mangoldt function, respectively. Using this estimate, we relate the Goldbach problem to the problem of showing that for all $N>6$ $(N\neq 2p)$, If $\sum \limits_{p|N}\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0$, then $\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0$ for each prime $p|N$.
Category: Number Theory
[16] viXra:2007.0116 [pdf] submitted on 2020-07-15 00:59:51
Authors: Dmitri Martila
Comments: 7 Pages. Rejected by many top journals without review
There are tens of self-proclaimed proofs for the Riemann Hypothesis and only 2 or 4 disproofs of it in arXiv. I am adding to the Status Quo my very short and clear results even without explicit mentioning of the prime numbers. One of my breakthroughs uses the peer-reviewed achievement of Dr.Sole and Dr.Zhu, published just 4 years ago in a serious mathematical journal INTEGERS.
Category: Number Theory
[15] viXra:2007.0115 [pdf] replaced on 2021-07-27 18:49:17
Authors: Dmitri Martila
Comments: 5 Pages.
I derive a new equivalent formulation of Goldbach's strong conjecture and present several proofs of Goldbach's strong conjecture and other conjectures. You are free not to get enlightened about that facts. But please pay respect to new dispositions of the conjectures and research methods in this note.
Category: Number Theory
[14] viXra:2007.0114 [pdf] submitted on 2020-07-15 01:04:48
Authors: Dmitri Martila
Comments: 5 Pages. Rejected by many top journals without review
Relying on the validity of Dr.Dahmen's peer-reviewed result, I am refuting the abc-conjecture even without explicit mentioning prime numbers.
Category: Number Theory
[13] viXra:2007.0105 [pdf] replaced on 2020-11-20 21:11:58
Authors: Dante Servi
Comments: 42 Pages. Copyright by Dante Servi. With this revision I update "summary" and the "conclusion" both at the end of Appendix 1.
The prime numbers have a distribution that is only apparently random, with this article I will demonstrate that the distribution derives from the combination of the sequences of the various prime numbers, giving a demonstration that I define as graphic. I trust that this demonstration will prove the validity or otherwise of Riemann's hypothesis (I believe in validity).
Category: Number Theory
[12] viXra:2007.0102 [pdf] submitted on 2020-07-14 13:49:14
Authors: Dhananjay Phatak
Comments: 79 Pages.
In the literature [1],
Carmichael Numbers that satisfy additional
constraints $(p+1) \mydivides (N+1)$ for every prime divisor
$p \mydivides N$ are referred to as
``Williams' Numbers''\footnote{more precisely, ``1-Williams Numbers''~;
however~; the distinctions between different types of
Willliams' numbers are not relevant in this document and therefore,
we refer to 1-Williams Numbers~ simply as Williams' numbers.}.
%
In the renowned Pomerance-recipe~\cite{pomerance1984there}
to search for Baillie-PSW pseudoprimes; there are
heuristic arguments suggesting that the number of Williams'
Numbers could be large (or even unlimited). Moreover,
it is shown~\cite{pomerance1984there}
that if a Williams' number
is encountered during a search in accordance with all of the
conditions
in that recipe~\cite{pomerance1984there}~;
then it must also be a Baillie-PSW pseudoprime.
We derive new analytic bounds on the prime-divisors of
a Williams' Number.\\
Application of the bounds to
Grantham's set of 2030 primes~(see ~\cite{grantham-620-list})
drastically reduces the search
space from the impossible size $\approx 2^{(2030)}$ to less than
a quarter billion cases (160,681,183 cases to be exact, please
see the appendix for details).
We tested every single case in the reduced
search space with maple code. The result showed that
there is \underline{NO Williams' number (and therefore NO Baillie-PSW
pseudo-prime which is also a Williams' number)} in the entire space of
subsets of the Grantham-set. The results thus demonstrate that
Williams' numbers either do not exist or are extremely rare.
We believe the former; i.e., that No such
composite (i.e., a Williams' Number of this type) exists.
Category: Number Theory
[11] viXra:2007.0095 [pdf] submitted on 2020-07-14 11:02:16
Authors: Florent Raynal
Comments: 3 Pages. French writing
Syracuse theory demonstration in French.
Category: Number Theory
[10] viXra:2007.0090 [pdf] submitted on 2020-07-13 17:52:48
Authors: Mohammed zohal
Comments: 14 Pages.
In the letter sent by Goldbach to Euler in 1742 (Christian, 1742) he stated that its seems that
every odd number greater than 2 can be expressed as the sum of three primes. As reformulated
by Euler, an equivalent form of this conjecture called the strong or binary Goldbach conjecture
states that all positive even integers greater or equal to 4 can be expressed as the sum of two primes which are sometimes called a Goldbach partition. Jorg (2000) and Matti (1993) have verified it
up to 4.1014. Chen (1973) has shown that all large enough even numbers are the sum of a prime
and the product of at most two primes... The majority of mathematicians believe that Goldbach's conjecture is true, especially on statistical
considerations ,on the subject we give the proof of Goldbach's strong conjecture whose veracity is
based on a clear and simple approach.
Category: Number Theory
[9] viXra:2007.0086 [pdf] submitted on 2020-07-13 20:59:01
Authors: Theophilus Agama
Comments: 6 Pages.
Using some properties of the prime, we establish an estimate for the sum \begin{align}\sum \limits_{k\geq 2}\bigg(\frac{1}{2}+o(1)\bigg)\int \limits_{2}^{x}\frac{\pi_k(t)}{t}dt=\frac{x}{2}+O\bigg(\frac{x}{\log x}\bigg).\nonumber
\end{align}
Category: Number Theory
[8] viXra:2007.0071 [pdf] submitted on 2020-07-12 12:21:17
Authors: Pedro Hugo García Peláez
Comments: 2 Pages.
We can find all prime numbers in steps of Fibonacci or Lucas numbers.
Category: Number Theory
[7] viXra:2007.0067 [pdf] submitted on 2020-07-11 19:35:45
Authors: Pedro Hugo García Peláez
Comments: 4 Pages.
We can find infinite prime numbers with the separation we want and we can express every even number as the sum of two prime numbers.
Category: Number Theory
[6] viXra:2007.0059 [pdf] replaced on 2020-07-10 11:18:52
Authors: Fabrizio Vassallo
Comments: 1 Page.
The infinite sum of a "fractal" set of numbers is found. The result is intended as an example of recreational mathematics, so we don’t worry about mathematical rigor.
Category: Number Theory
[5] viXra:2007.0042 [pdf] replaced on 2020-07-07 15:25:58
Authors: Roberto Amato
Comments: 5 Pages. Preprint of Paper accepted for publication in International Journal of Mathematics and Computer Science, Volume 16, no. 1, 2021, 143–147.
Some relations among Pythagorean triples are established. The main tool is a fundamental characterization of the Pythagorean triples through a cathetus that allows to determine the relationships between two Pythagorean triples with an assigned cathetus a and b and the Pythagorean triple with cathetus a · b.
Category: Number Theory
[4] viXra:2007.0038 [pdf] replaced on 2020-10-30 04:49:14
Authors: Jorma Jormakka
Comments: 24 Pages. This is a abridged version of the paper, to be submitted very soon.
This paper proves that the Birch and Swinnerton-Dyer conjecture fails in rank one, unlike is claimed in the CMI problem statement.
Category: Number Theory
[3] viXra:2007.0013 [pdf] submitted on 2020-07-02 19:50:29
Authors: Xuan Zhong Ni
Comments: 5 Pages.
In this article, we use method of a modified sieve of Eratosthenes to prove the Goldbach Conjecture.
Category: Number Theory
[2] viXra:2007.0011 [pdf] replaced on 2020-07-22 11:42:36
Authors: R. Rama Chander
Comments: 21 Pages. This paper proposed three novel and unique methods to achieve the much desired Integer multiplication in time O(n).
This paper attempts to disprove the asymptotic in time O(n log n) prediction of Schönhage-Strassen, claiming its ‘best possible’ result and remarking that no one will ever find a faster multiplication algorithm. Accordingly, this paper postulates that, the most desired complexity in time, i.e. O(n) is as achievable using only two basic arithmetic operations.
We present four algorithms for large integer multiplications. First algorithm is based on the place value approach and achieves the much desired complexity of O(n), based on Nearest Place Values (NPV) approach. Second algorithm extends and improves Karatsuba algorithm for any ordered pairs greater than 2. It is important to remind that the present version of Karatsuba algorithm works only for ordered pairs of 2. Third algorithm called Addition and Subtraction (AnS) achieves time complexity of O(n) for very large integer multiplications using only repeated additions and subtractions. The fourth algorithm is called the Repeated Doubling Method (RDM), which is an improvised version of AnS algorithm and achieves time complexity of O(n).
Category: Number Theory
[1] viXra:2007.0005 [pdf] submitted on 2020-07-01 11:29:17
Authors: Aaron Chau
Comments: 3 Pages.
实际上寻找质数与函数根本无关, 比如在西方的古希腊, Euclid证明质数无限, 他是用(乘除法)来表述反证法；而现时在东方香港，本文同时来证明孪生质数无限, 黎曼假设不成立；筆者是用(加减法来表述多与少)是永恒。
Category: Number Theory
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