scholarly journals Proof the Skewes’ number is not an integer using lattice points and tangent line

2021 ◽  
Vol 17 (2) ◽  
pp. 5-18
Author(s):  
V. Ďuriš ◽  
T. Šumný ◽  
T. Lengyelfalusy
Keyword(s):  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Integral Function ◽  
Lattice Points ◽  
Tangent Line ◽  
Miller Test ◽  
Logarithmic Integral ◽  
The Difference ◽  
Prime Counting Function

Abstract Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

scholarly journals Combinatorial Models of the Distribution of Prime Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1224
Author(s):  
Vito Barbarani
Keyword(s):  
Prime Number ◽  
Counting Function ◽  
Stirling Numbers ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Lower And Upper Bounds ◽  
Integer Sequences ◽  
New Class ◽  
Random Integer ◽  
Prime Counting Function

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

scholarly journals Combinatorial Models of the Distribution of Prime Numbers

2021 ◽  
Author(s):  
Vito Barbarani
Keyword(s):  
Prime Number ◽  
Counting Function ◽  
Stirling Numbers ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Lower And Upper Bounds ◽  
Integer Sequences ◽  
New Class ◽  
Random Integer ◽  
Prime Counting Function

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

scholarly journals The Elementary Proof of the Riemann's Hypothesis

2020 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Mathematical Theory ◽  
Elementary Proof ◽  
Counting Function ◽  
Binomial Coefficients ◽  
Complex Issue ◽  
Logarithmic Integral ◽  
Prime Counting Function ◽  
Prime Gaps ◽  
The Relationship ◽  
Very High

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

scholarly journals The Elementary Proof of the Riemann's Hypothesis

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Mathematical Theory ◽  
Elementary Proof ◽  
Counting Function ◽  
Binomial Coefficients ◽  
Complex Issue ◽  
Logarithmic Integral ◽  
Prime Counting Function ◽  
Prime Gaps ◽  
The Relationship ◽  
Very High

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

New consequences of prime-counting function

2021 ◽  
Vol 27 (4) ◽  
pp. 25-31
Author(s):  
Sadani Idir ◽  
Keyword(s):  
Counting Function ◽  
Prime Numbers ◽  
Prime Counting Function

Our objective in this paper is to study a particular set of prime numbers, namely \ left \ {p \ in \ mathbb {P} \ \ text {and} \ \ pi (p) \ notin \ mathbb {P} \ right \} \ !.As a consequence, estimations of the form \ sum {f (p)}with p being prime belonging to this set are derived.

scholarly journals An Investigation on the Prime and Twin Prime Number Functions by Periodical Binary Sequences and Symmetrical Runs in a Modified Sieve Procedure

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 775
Author(s):  
Bruno Aiazzi ◽  
Stefano Baronti ◽  
Leonardo Santurri ◽  
Massimo Selva
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Integral Function ◽  
Point Of View ◽  
Binary Sequences ◽  
Prime Characteristic ◽  
Twin Primes ◽  
Logarithmic Integral ◽  
The One ◽  
Prime Function

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to the case of twin primes is formulated. The proposed investigation, which is named Limited INtervals into PEriodical Sequences (LINPES) relies on a set of binary periodical sequences that are evaluated in limited intervals of the prime characteristic function. These sequences are built by considering the ensemble of deleted (that is, 0) and undeleted (that is, 1) integers in a modified version of the Sieve procedure, in such a way a symmetric succession of runs of zeroes is found in correspondence of the gaps between the undeleted integers in each period. Such a formulation is able to estimate the prime number function in an equivalent way to the logarithmic integral function Li(x). The present analysis is then extended to the twin primes, by taking into account only the runs whose size is two. In this case, the proposed procedure gives an estimation of the twin prime function that is equivalent to the one of the logarithmic integral function Li 2 ( x ) . As a consequence, a possibility is investigated in order to count the twin primes in the same intervals found for the primes. Being that the bounds of these intervals are given by squares of primes, if such an inference were actually proved, then the twin primes could be estimated up to infinity, by strengthening the conjecture of their never-ending.

Chebysev’s Estimate of the Prime Counting Function

2020 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Prime Number ◽  
Historical Perspective ◽  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Counting Function

In this article we shall discuss about Chebyshev’s estimates of the prime counting function, which was later superseded by the Prime Number Theorem, nonetheless it is significant from both mathematical and historical perspective. Chebyshev’s estimates of the prime counting function forms the basis and motivation for the Prime number theorem derived later by mathematicians.

Chebysev’s Estimate of the Prime Counting Function

2020 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Prime Number ◽  
Historical Perspective ◽  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Counting Function

In this article we shall discuss about Chebyshev’s estimates of the prime counting function, which was later superseded by the Prime Number Theorem, nonetheless it is significant from both mathematical and historical perspective. Chebyshev’s estimates of the prime counting function forms the basis and motivation for the Prime number theorem derived later by mathematicians.

scholarly journals Quantum computation of prime number functions

2014 ◽  
Vol 14 (7&8) ◽  
pp. 577-588
Author(s):  
Jose I. Latorre ◽  
German Sierra
Keyword(s):  
Counting Function ◽  
Quantum Circuit ◽  
Prime Numbers ◽  
Quantum Systems ◽  
Quantum Superposition ◽  
Arithmetic Properties ◽  
Twin Primes ◽  
Primality Test ◽  
Entanglement Measures ◽  
Prime Counting Function

We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than $2^n$, where $n$ is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.

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