[26] **viXra:1703.0243 [pdf]**
*replaced on 2017-03-29 16:58:21*

**Authors:** Wes Hansen

**Comments:** 9 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, in turn, enables our demonstration of a counter-example to Tennenbaum’s Theorem.

**Category:** Number Theory

[25] **viXra:1703.0241 [pdf]**
*submitted on 2017-03-26 03:17:22*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt is essentially an inductive approach.

**Category:** Number Theory

[24] **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

[23] **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

[22] **viXra:1703.0220 [pdf]**
*replaced on 2017-03-29 20:38:48*

**Authors:** Pedro Caceres

**Comments:** 23 Pages.

Abstract: 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.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behavior of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
The way to build the sequence of prime numbers uses sieves, an algorithm yielding all primes up to a given limit, using only trial division method which consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime.
This paper introduces a new way to approach prime numbers, called the DNA-prime structure because of its intertwined nature, and a new process to create the sequence of primes without direct division or multiplication, which will allow us to introduce a new primality test, and a new factorization algorithm.
As a consequence of the DNA-prime structure, we will be able to provide a potential proof of Golbach’s conjecture.

**Category:** Number Theory

[21] **viXra:1703.0211 [pdf]**
*submitted on 2017-03-22 01:05:49*

**Authors:** Simon Plouffe

**Comments:** 41 Pages. Conference is in French

Une conférence sur Pi, le jour de Pi : 14 mars 2017 au Lycée International Winston Churchill : Londres.
A conference on Pi on Pi Day, march 14 2017 at the Winston Churchill International College (Lycée ) London.

**Category:** Number Theory

[20] **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

[19] **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

[18] **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

[17] **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

[16] **viXra:1703.0155 [pdf]**
*submitted on 2017-03-15 23:40:37*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

Near m, the destance of primes is lower order than logm. This is
the key to solve the Legendre's conjecture.

**Category:** Number Theory

[15] **viXra:1703.0147 [pdf]**
*replaced on 2017-03-23 15:17:52*

**Authors:** Philip A. Bloom

**Comments:** 2 Pages.

This two-page proof, by contraposition, of Fermat's last theorem, uses two analogous forms of a previously overlooked equation, each of which is equivalent to rational z^n-y^n = x^n. A relationship held in common by these two equations reveals the crucial n = 1, 2 limitation.

**Category:** Number Theory

[14] **viXra:1703.0124 [pdf]**
*submitted on 2017-03-13 13:55:36*

**Authors:** Petr E. Pushkarev

**Comments:** 5 Pages.

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

[13] **viXra:1703.0115 [pdf]**
*replaced on 2017-03-17 20:27:33*

**Authors:** John Yuk Ching Ting

**Comments:** 18 Pages. This research paper contains the rigorous proofs for Polignac's and Twin prime conjectures.

Prime numbers and composite numbers are intimately related simply because the complementary set of composite numbers constitutes the set of natural numbers with the exact set of prime numbers excluded in its entirety. In this research paper, we use our 'Virtual container' method, which predominantly incorporates the novel mathematical tool coined Information-Complexity conservation with its core foundation based on this [complete] prime-composite number relationship, to solve the intractable open problem of whether prime gaps are infinite (arbitrarily large) in magnitude with each individual prime gap generating prime numbers which are again infinite in magnitude. This equates to solving Polignac's conjecture which involves analysis of all possible prime gaps = 2, 4, 6,... and [the subset] Twin prime conjecture which involves analysis of prime gap = 2 (for twin primes). In conjunction with our cross-referenced 2017-dated research paper entitled "Rigorous proof for Riemann hypothesis using Sigma-Power Laws" http://viXra.org/abs/1703.0114, we advocate for our ambition that the Virtual container method be considered as a new method of mathematical proof especially for ’Special-Class-of-Mathematical-Problems with Solitary-Proof-Solution’.

**Category:** Number Theory

[12] **viXra:1703.0114 [pdf]**
*replaced on 2017-03-18 01:01:54*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. This research paper contains the rigorous proof for Riemann hypothesis.

The triple countable infinite sets of (i) x-axis intercepts, (ii) y-axis intercepts, and (iii) both x- and y-axes [formally known as the 'Origin'] intercepts in Riemann zeta function are intimately related to each other simply because they all constitute complementary points of intersection arising from the single [exact same] countable infinite set of curves generated by this function. This [complete] relationship amongst all three sets of intercepts will enable us to simultaneously study important intrinsic properties derived from all those intercepts in a mathematically consistent manner which then provides the rigorous proof for Riemann hypothesis as well as fully explain x-axis intercepts (which is the usual traditionally-dubbed 'Gram points') and y-axis intercepts. Riemann hypothesis involves analysis of all nontrivial zeros of Riemann zeta function and refers to the celebrated proposal by famous German mathematician Bernhard Riemann in 1859 whereby all nontrivial zeros are conjectured to be located on the critical line [or equivalently stated as all nontrivial zeros are conjectured to exactly match the Origin intercepts]. Concepts from the Hybrid method of Integer Sequence classification, together with our key formulae coined Sigma-Power Laws, are some of the important mathematical tools employed in this paper to successfully achieve our proof. Not least in [again] using the same 'Virtual container' method in this current research paper, there are other additional deeply inseparable mathematical connections between the content of this paper and our 2017-dated publication on the dual source of prime number infiniteness entitled "Rigorous proofs for Polignac's and Twin prime conjectures using Information-Complexity conservation" http://viXra.org/abs/1703.0115.

**Category:** Number Theory

[11] **viXra:1703.0104 [pdf]**
*submitted on 2017-03-11 10:45:33*

**Authors:** Pedro Caceres

**Comments:** 25 Pages.

PrimeNumbers 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 advance 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 this 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:
(1)ζ(z) = ∑1/n^z
The Zeta function, 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 these zeros, 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 non-trivial zero z* of the Riemann Zeta function is 1/2.
Thus, if the hypothesis is correct, all the non-trivial zeros lie on the critical line consisting of the complex numbers 1/2 + i ß, where ß is a real number and i is the imaginary unit.
In this paper, we will analyze the Riemann Zeta function and provide an analytical/geometrical proof of the Riemann Hypothesis. The proof will be based on the fact that if we decompose the ζ(z) in a difference of two functions, both functions need to be equal when ζ(z)=0, so their distance to the origin or modulus must be equal and we will prove that this can only happen when Re(z)=1/2 for certain values of Im(z).
We will also prove that all non-trivial zeros of ζ(z) in the form z=1/2+iß have all ß related by an algebraic expression. They are all connected and not independent.
Finally, we will show that as a consequence of this connection of all ß, the harmonic function Hn can be expressed as a function of each ß zero of ζ(z) with infinite representations.
We will use mathematical and computational methods available for engineers.

**Category:** Number Theory

[10] **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

[9] **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

[8] **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

[7] **viXra:1703.0072 [pdf]**
*submitted on 2017-03-07 22:53:09*

**Authors:** Kolosov Petro

**Comments:** 12 Pages. Final revised paper, text overlap with previous versions

This paper presents the way to make expansion for the next form function: y =
x n , ∀(x,n) ∈ N to the numerical series. The most widely used methods to solve this
problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that
is, derivative and integral are inverse operators). The paper provides the other kind
of solution, based on induction from particular to general case, except above described
theorems.

**Category:** Number Theory

[6] **viXra:1703.0048 [pdf]**
*replaced on 2017-03-14 18:11:01*

**Authors:** Stephen Crowley

**Comments:** 6 Pages.

Abstract. It is conjectured that when t=t_n is the imaginary part of the n-th zero of ζ on the critical line, the normalised argument S(t_)_=π^(-1)argζ(1/2+i t__) is equal to S(t)=S_n(t_n)=_n-3/2-(ϑ(t_n_))/π where ϑ(t) is the Riemann-Siegel ϑ function. If S(t_n)=S_n(t_n)∀n∈ℤ^+ then the exact transcendental equation for the Riemann zeros has a solution for each positive integer n which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip in that case.

**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