Positive approach: Implications for the relation between number theory and geometry, including connection to Santilli mathematics, from Fibonacci reconstitution of natural numbers and of prime numbers

2014 ◽  
Author(s):  
Stein E. Johansen
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Positive Approach

scholarly journals Some interesting tasks from the classical number theory

2020 ◽  
pp. 150-188
Author(s):  
Zurab Agdgomelashvili ◽  
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Prime Divisors ◽  
Problem Class

The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q is an odd prime number, and p N \{1}, p]1;q] [q  2; 2q] , then each of the different prime divisors of the number 1 1 pq A p    taken separately is greater than p; Theorem 4. Let’s q is an odd prime number, and p{q1; 2q1}, then from different prime divisors of the number 1 1 pq A p    taken separately, only one of them is less than p. A has at least two different simple divisors. Task 1. Solve the equation 1 2 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 2. Solve the equation 1 3 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 3. Solve the equation 1 1 z x y p y    where p{6; 7; 11; 13;} are the prime numbers, x, y  N and y is a prime number. There is a lema with which the problem class can be easily solved: Lemma ●. Let’s a, b, nN and (a,b) 1. Let’s prove that if an  0 (mod| ab|) , or bn  0 (mod| ab|) , then | ab|1. Let’s solve the equations ( – ) in natural x , y numbers: I. 2 z x y z z x y          ; VI. (x  y)xy  x y ; II. (x  y)z  (2x)z  yz ; VII. (x  y)xy  yx ; III. (x  y)z  (3x)z  yz ; VIII. (x  y) y  (x  y)x , (x  y) ; IV. ( y  x)x y  x y , (y  x) ; IX. (x  y)x y  xxy ; V. ( y  x)x y  yx , (y  x) ; X. (x  y)xy  (x  y)x , (y  x) . Theorem . If a, bN (a,b) 1, then each of the divisors (a2  ab  b2 ) will be similar. The concept of pseudofibonacci numbers is introduced and some of their properties are found.

Distribution of adjacent prime numbers in the Ulam spiral

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Vicente Jara-Vera ◽  
Carmen Sánchez-Ávila
Keyword(s):  
Number Theory ◽  
Design Methodology ◽  
Prime Numbers ◽  
Research Interest ◽  
Natural Numbers ◽  
Content Type ◽  
Cryptographic System ◽  
Approximate Expressions

Purpose The distribution of natural numbers in the Ulam spiral manifests a series of unexpected regularities of the elusive prime numbers. Their sequencing remains a topic of research interest, with many ramifications in different branches of Mathematics, especially in number theory and the prime factorisation problem. Accordingly, the focus of the research is on the most known and widespread asymmetric cryptographic system, that is, the RSA encryption. Design/methodology/approach This paper presents the presence of one, two, three or four adjacencies for the diverse primes that appear in a spiral, considering the Hardy–Littlewood twin prime conjecture and the constellations of primes. Findings Through equations, the calculation formulas of primes inside a spiral that have one to four primes in their adjacent places is offered, with approximate expressions that facilitate the operations, showing that the adjacencies decrease very rapidly as the spiral progresses, although without disappearing. Originality/value A generalisation to cases in which the distances to the prime values change in an ascending way in accordance with the step of the Ulam spiral is offered.

The diophantine equation x2 + paqb = yq

2021 ◽  
pp. 1-18
Author(s):  
Hemar Godinho ◽  
Victor G. L. Neumann
Keyword(s):  
Diophantine Equation ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Lucas Sequences ◽  
Primitive Divisors

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

Some comments on inverse arithmetic functions

2004 ◽  
Vol 89 (516) ◽  
pp. 403-408
Author(s):  
P. G. Brown
Keyword(s):  
Number Theory ◽  
Prime Factor ◽  
Arithmetic Functions ◽  
Natural Numbers ◽  
Finite Mathematics ◽  
Factor Decomposition

In many of the basic courses in Number Theory, Finite Mathematics and Cryptography we come across the so-called arithmetic functions such as ϕn), σ(n), τ(n), μ(n), etc, whose domain is the set of natural numbers. These functions are well known and evaluated through the prime factor decomposition of n. It is less well known that these functions possess inverses (with respect to Dirichlet multiplication) which have interesting properties and applications.

scholarly journals About the proof-theoretic ordinals of weak fixed point theories

1992 ◽  
Vol 57 (3) ◽  
pp. 1108-1119 ◽  
Author(s):  
Gerhard Jäger ◽  
Barbara Primo
Keyword(s):  
Fixed Point ◽  
Number Theory ◽  
Second Order ◽  
Order Number ◽  
Natural Numbers

AbstractThis paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.

scholarly journals SOME REMARKS ON PERMUTATIONS WHICH PRESERVE THE WEIGHTED DENSITY

2013 ◽  
Vol 56 (1) ◽  
pp. 35-45
Author(s):  
Milan Paštéka ◽  
Zuzana Václavíková
Keyword(s):  
Number Theory ◽  
Natural Numbers ◽  
Logarithmic Density ◽  
Weighted Density

ABSTRACT In this paper we study the conditions (1), (2) and (3) for the permutations which preserve the weighted density. These conditions are motivated by the conditions of Lévy group, originated in [Levy, P.: Problèmes concrets d’Analyse Fonctionelle. Gauthier Villars, Paris, 1951], and studied in [Obata, N.: Density of natural numbers and Lévy group, J. Number Theory 30 (1988), 288-297]. In the second part we prove that under some conditions for the sequence of weights, for instance for the logarithmic density, the first two conditions can be launched

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

A derivation of number theory from ancestral theory

1952 ◽  
Vol 17 (3) ◽  
pp. 192-197 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Functional Calculus ◽  
Natural Numbers ◽  
Class Inclusion ◽  
First Order ◽  
Free Variables

Martin has shown that the notions of ancestral and class-inclusion are sufficient to develop the theory of natural numbers in a system containing variables of only one type.The purpose of the present paper is to show that an analogous construction is possible in a system containing, beyond the quantificational level, only the ancestral and the ordered pair.The formulae of our system comprise quantificational schemata and anything which can be obtained therefrom by writing pairs (e.g. (x; y), ((x; y); (x; (y; y))) etc.) for free variables, or by writing ancestral abstracts (e.g. (*xyFxy) etc.) for schematic letters, or both.The ancestral abstract (*xyFxy) means what is usually meant by ; and the formula (*xyFxy)zw answers to Martin's (zw; xy)(Fxy).The system presupposes a non-simple applied functional calculus of the first order, with a rule of substitution for predicate letters; over and above this it has three axioms for the ancestral and two for the ordered pair.

scholarly journals III. The sums of the series of the reciprocals of the prime numbers and of their powers

1882 ◽  
Vol 33 (216-219) ◽  
pp. 4-10 ◽  
Keyword(s):  
Prime Numbers ◽  
Natural Numbers ◽  
The Difference

Euler has shown that it is possible to sum the series of reciprocals of powers of the prime numbers, and he has calculated the values of these sums for the even powers. I thought it of some interest to calculate the sums for the odd powers, and to evaluate a peculiar constant (somewhat analogous to the Eulerian constant,— γ = 0·57721 56649 01532 86060 65) which presents itself, in the series of simple reciprocals of primes, as the difference between the sum of the series and the double logarithmic infinity to the Napierian base ϵ. The summation of these series was shown by Euler to depend upon the Napierian logarithms of the sums of the reciprocals of the powers of the natural numbers.

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