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[21] viXra:1703.0304 [pdf] replaced on 2021-07-15 20:35:36
Authors: Abdelmajid Ben Hadj Salem
Comments: 9 Pages. Submitted to the Arabian Journal of Mathematics. Comments welcome.
In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) $s=\sigma+it$ of the zeta function, defined by:
$$\zeta(s) = \sum_{n=1}^{+\infty}\frac{1}{n^s},\,\mbox{for}\quad \Re(s)>1$$
have real part $\sigma= 1/2$.
We give a proof that $\sigma= 1/2$ using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet $\eta$ function.
Category: Number Theory
[20] viXra:1703.0297 [pdf] submitted on 2017-03-31 05:36:04
Authors: Marius Coman
Comments: 2 Pages.
In this paper I note two sequences of Poulet numbers: the terms of the first sequence are the Poulet numbers which can be written as P*2 – d; the terms of the second sequence are the Poulet numbers which can be written as (P*2 – d)*2 - d, where P is another Poulet number and d one of the prime factors of P. I also conjecture that the both sequences are infinite and I observe that the recurrent relation ((((P*2 – d)*2 – d)*2 – d)...) conducts sometimes to more than one Poulet number (for instance, starting with P = 4369 and d = 257, the first, the second and the third numbers obtained are 8481, 16705 and 33153, all three Poulet numbers).
Category: Number Theory
[19] viXra:1703.0243 [pdf] replaced on 2018-06-12 14:12:55
Authors: Wes Hansen
Comments: 11 Pages.
In what follows we develop foundations for a set of non-standard natural numbers we call q-naturals, where q stands for quanta, by the recursive generation of reflexive sets. From the practical perspective, these q-naturals correspond to ordered pairs of natural numbers with the lexicographic ordering, hence, they are isomorphic to ω^2. In addition, we demonstrate a novel definition of the arithmetical operation, multiplication, which turns out to be recursive. This operation, together with lexicographic order and coordinate-wise addition, defines an arithmetical structure which extends the “standard” model but yet has a recursive order relation and recursive arithmetical operations defined on the entire domain.
Category: Number Theory
[18] viXra:1703.0241 [pdf] replaced on 2017-10-19 07:27:00
Authors: Barry Foster
Comments: 2 Pages.
Goldbach Conjecture This attempt requires very little mathematical knowledge.
Category: Number Theory
[17] viXra:1703.0237 [pdf] submitted on 2017-03-25 08:25:21
Authors: Ricardo Gil
Comments: 4 Pages.
The purpose of this papers is to share an encryption system based on a modified Riemann Zeta function which relates to prime
numbers.
Category: Number Theory
[16] viXra:1703.0226 [pdf] submitted on 2017-03-23 22:58:58
Authors: Ramesh Chandra Bagadi
Comments: 4 Pages.
In this research investigation, the author has detailed about the Scheme of construction of Natural metric for any given positive Integer. Natural Metric can be used for Natural Scaling of any Set optimally. Natural Metric also forms the Universal Basis for the Universal Correspondence Principle between Quantum mechanics and Newtonian Mechanics. Furthermore, Natural Metric finds great use in the Science of Forecasting Engineering.
Category: Number Theory
[15] viXra:1703.0220 [pdf] replaced on 2020-05-03 18:43:46
Authors: Pedro Jesus Caceres
Comments: 49 Pages.
A Prime number (or a Prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The crucial importance of Prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, which states that every integer larger than 1 can be written as a product of one or more Primes in a way that is unique except for the order of the Prime factors. Primes can thus be considered the “basic building blocks”, the atoms, of the natural numbers.
In this paper we present 30 ideas about Primes. Some are based on the fact that all Primes greater than 3, are 1 unit away from a multiple of 1, 2, 3, 4, or 6, which is used to introduce new methods to factorize, to count Primes less than a given number, and to add some ideas to already famous Prime conjectures.
Category: Number Theory
[14] viXra:1703.0192 [pdf] submitted on 2017-03-20 08:06:50
Authors: Helmut Preininger
Comments: 10 Pages.
This paper introduces the notion of an S-Structure (short for Squarefree Structure.) After establishing a few simple properties of such S-Structures, we investigate the squarefree natural numbers as a primary example. In this subset of natural numbers we consider "arithmetic" sequences with varying initial elements. It turns out that these sequences are always periodic. We will give an upper bound for the minimal and maximal points of these periods.
Category: Number Theory
[13] viXra:1703.0180 [pdf] submitted on 2017-03-19 02:37:08
Authors: Marius Coman
Comments: 1 Page.
In this paper I conjecture that any number of the form 4*n^2 + 8*n + 3, where n is positive integer, is Fermat pseudoprime to base 2*n + 2.
Category: Number Theory
[12] viXra:1703.0177 [pdf] submitted on 2017-03-18 07:50:11
Authors: Marius Coman
Comments: 1 Page.
In this paper I conjecture that any Poulet number of the form (4^n + 1)/5 where n is prime is either 2-Poulet number either a product of primes p(1)*p(2)*...*p(k) such that all the semiprimes p(i)*p(j), where 1 ≤ i < j ≤ k, are 2-Poulet numbers.
Category: Number Theory
[11] viXra:1703.0174 [pdf] submitted on 2017-03-18 04:21:30
Authors: Marius Coman
Comments: 2 Pages.
In this paper I conjecture that any number of the form (4^n – 1)/3 where n is odd greater than 3 is divisible by a Poulet number (it is known that any number of this form is a Poulet number if n is prime greater than 3; such a number is called Cipolla pseudoprime to base 2, see the sequence A210454 in OEIS).
Category: Number Theory
[10] viXra:1703.0124 [pdf] submitted on 2017-03-13 13:55:36
Authors: Petr E. Pushkarev
Comments: 5 Pages. was published in the Global Journal of Pure and Applied Mathematics 13, no. 6 (2017): 1987-1992
In this article we are closely examining Riemann zeta function's non-trivial zeros. Especially, we examine real part of non-trivial zeros. Real part of Riemann zeta function's non-trivial zeros is considered in the light of constant quality of such zeros. We propose and prove a theorem of this quality. We also uncover a definition phenomenons of zeta and Riemann xi functions. In conclusion and as an conclusion we observe Riemann hypothesis in perspective of our researches.
Category: Number Theory
[9] viXra:1703.0104 [pdf] replaced on 2020-03-30 22:34:50
Authors: Pedro Jesus Caceres
Comments: 39 Pages.
Abstract: 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 a proof of the RH. The proof of the RH will be built following these five parts:
PART 1:Description of the RZF ζ(z)
PART 2: The C-transformation
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)
PART 4:
4.1. Analysis of the values of z such that X(z)=Y(z), and |X(z)|=|Y(z)|, that equates to ζ(z)=0
4.2. Proof that |X(z)|=|Y(z)| only if Re(z)=1/2
4.3. Conclude that ζ(z)=0 only if Re(z)=1/2 for Re(z)≥0
PART 5: We will also prove that all non-trivial zeros of ζ(z) in the critical line of the form z=1/2+ßi are not distributed randomly. There is a relationship between the values of those zeros and the Harmonic function that leads to an algebraic relationship between any two zeros.
We will use mathematical and computational methods available for engineers.
Category: Number Theory
[8] viXra:1703.0097 [pdf] submitted on 2017-03-11 02:01:57
Authors: Wu ShengPing
Comments: 4 Pages.
The main idea of this article is simply calculating integer
functions in module. The algebraic in the integer modules is studied in
completely new style. By a careful construction the result that
two finite numbers is with unequal logarithms in a corresponding module is proven, which result is applied to solving
a kind of diophantine equation: $c^q=a^p+b^p$.
Category: Number Theory
[7] viXra:1703.0086 [pdf] submitted on 2017-03-09 09:40:06
Authors: Stephen Marshall
Comments: 8 Pages.
This paper presents a complete and exhaustive proof of Landau's Fourth Problem. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m:
m = (p-1)!( 1/p + ((-1)^d (d!))/(p+d)) + 1/p + 1/(p+d)
We use this proof for d = 2n + 1 to prove the infinitude of Landau’s Fourth Problem prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Landau’s Fourth Problem possible.
Category: Number Theory
[6] viXra:1703.0078 [pdf] submitted on 2017-03-08 10:45:40
Authors: Wu ShengPing
Comments: 4 Pages.
The main idea of this article is simply calculating integer
functions in module. The algebraic in the integer modules is studied in
completely new style. By a careful construction a result is obtained on
two finite numbers with unequal logarithms, which result is applied to solving
a kind of diophantine equations.
Category: Number Theory
[5] viXra:1703.0040 [pdf] submitted on 2017-03-04 11:30:54
Authors: Antoine Balan
Comments: 3 pages, written in French
We propose in the present paper to consider the Riemann Hypothesis asympotically (ARH) ; it means when the imaginary part of the zero in the critical band is great. We show that the problem, expressed in these terms, is equivalent to the fact that an equation called the * equation has only a finite number of solutions, but we have not proved it.
Category: Number Theory
[4] viXra:1703.0033 [pdf] submitted on 2017-03-03 15:32:44
Authors: Reuven Tint
Comments: 5 Pages. original papper in russian
Keywords: three-term equation, the method of infinite growth, elementary aspect.
Annotation. An infinitely lifting method for making certain types of three-term equations, which is completely refuted by the ABC conjecture.
Category: Number Theory
[3] viXra:1703.0022 [pdf] submitted on 2017-03-03 10:27:01
Authors: Peter Bissonnet
Comments: 5 Pages.
This paper elucidates the major points of the above referenced paper.
1. Emphasizes the derivation of the double helices and that they are not arbitrarily chosen.
2. Explains why multiples of 42 appear in prime number theory.
3. Why s in 6s-1 and 6s+1 is really a composite number.
4. Why 2 and 3 are not true prime numbers based upon characteristics.
5. Philosophical reason as to the double helices falling more into a discoverable category (as in experimental physics), as opposed to being postulate driven.
Category: Number Theory
[2] viXra:1703.0021 [pdf] submitted on 2017-03-02 16:52:23
Authors: Stephen Marshall
Comments: 15 Pages.
This paper presents a complete proof of the Pell Primes are infinite. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer m:
m = (p-1)!( + ) + +
We use this proof for d = - to
prove the infinitude of Pell prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Pell Prime Conjecture possible.
Category: Number Theory
[1] viXra:1703.0005 [pdf] submitted on 2017-03-01 04:34:21
Authors: Ricardo Gil
Comments: 3 Pages.
The purpose of this paper is to provide algorithm that is 5 lines of code and that finds P & Q when N is given. It will work for RSA-2048 if the computer can float large numbers in PyCharm or Python. Also, the P&Q from Part I of the algorithm becomes the range for a for loop in Part II that returns and solves P*Q=N (True).
Category: Number Theory
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