Bounded Gaps Between Primes

2021 ◽  
Author(s):  
Kevin Broughan
Keyword(s):  
Number Theory ◽  
Graduate Students ◽  
Upper Bound ◽  
Finite Size ◽  
Computer Programs ◽  
Vast Array ◽  
Twin Primes ◽  
Background Material ◽  
Bounded Gaps

Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field.

scholarly journals Solving Two Problems IN Number Theory

2021 ◽  
Vol 2 (3) ◽  
pp. 8-9
Author(s):  
Mykhaylo Khusid
Keyword(s):  
Number Theory ◽  
Cause And Effect ◽  
Twin Primes

In 1995, Olivier Ramaret proved that any even number is the sum of no more than 6 primes. From the validity of Goldbach's ternary hypothesis (proved in 2013 year) it follows that any even number is the sum of not more than 4 numbers [1]. In the article, the author confirms the above and proves that the cause and effect of this is any even number the sum of not more than two prime and twin primes are infinite [8]-[14].

The Mordell Conjecture

2022 ◽  
Author(s):  
Hideaki Ikoma ◽  
Shu Kawaguchi ◽  
Atsushi Moriwaki
Keyword(s):  
Number Theory ◽  
Algebraic Geometry ◽  
Graduate Students ◽  
Algebraic Number ◽  
Algebraic Curve ◽  
Teaching Experience ◽  
Algebraic Number Theory ◽  
Diophantine Geometry ◽  
Weil Theorem ◽  
Mordell Conjecture

The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell–Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole.

Applied Computational Physics

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric. S. Swanson
Keyword(s):  
Real Life ◽  
Numerical Algorithms ◽  
Computer Programs ◽  
Computational Physics ◽  
Quantum Mechanical ◽  
Computational Techniques ◽  
Graduate Level ◽  
Problem Domain ◽  
Vast Array ◽  
Programming Skills

Applied Computational Physics describes methods for solving a vast array of classical and quantum mechanical scientific problems while stressing modern computational paradigms for achieving these solutions. The text develops computational techniques, numerical algorithms, and physics applications in parallel. The goal of the book is to provide students of physics with essential and modern computational skills and to increase the confidence with which they write computer programs within their problem domain. Hundreds of original problems reinforce programming skills and increase the ability to solve real-life physics problems at and beyond the graduate level.

scholarly journals R-prime numbers of degree k

2018 ◽  
Vol 38 (2) ◽  
pp. 75-82
Author(s):  
Abdelhakim Chillali
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Complexity Theory ◽  
Prime Numbers ◽  
Prime Pair ◽  
Random Input ◽  
Trap Door ◽  
Twin Primes ◽  
Open Questions ◽  
One Way Function

In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient of a function for it to be called one-way (see Theoretical Definition, in article). A twin prime is a prime number that has a prime gap of two, in other words, differs from another prime number by two, for example the twin prime pair (5,3). The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes p such that p + 2 is also prime. In this work we define a new notion: ‘r-prime number of degree k’ and   we give a new RSA trap-door one-way. This notion generalized a twin prime numbers because the twin prime numbers are 2-prime numbers of degree 1.

Questions of decidability and undecidability in Number Theory

1994 ◽  
Vol 59 (2) ◽  
pp. 353-371 ◽  
Author(s):  
B. Mazur
Keyword(s):  
Number Theory ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Computable Function ◽  
Integral Solution ◽  
Survey Article ◽  
Image Position ◽  
Parameter Values ◽  
Computable Functions ◽  
Integral Solutions

Davis, Matijasevic, and Robinson, in their admirable survey article [D-M-R], interpret the negative solution of Hilbert's Tenth Problem as a resounding positive statement about the versatility of Diophantine equations (that any listable set can be coded as the set of parameter values for which a suitable polynomial possesses integral solutions).One can also view the Matijasevic result as implying that there are families of Diophantine equations parametrized by a variable t, which have integral solutions for some integral values t = a > 0, and yet there is no computable function of t which provides an upper bound for the smallest integral solution for these values a. The smallest integral solutions of the Diophantine equation for these values are, at least sporadically, too large to be bounded by any computable function. This is somewhat difficult to visualize, since there is quite an array of computable functions. But let us take an explicit example. Consider the functionMatijasevic's result guarantees the existence of parametrized families of Diophantine equations such that even this function fails to yield an upper bound for its smallest integral solutions (for all values of the parameter t for which there are integral solutions).Families of Diophantine equations in a parameter t, whose integral solutions for t = 1, 2, 3,… exhibit a certain arythmia in terms of their size, have fascinated mathematicians for centuries, and this phenomenon (the size of smallest integral solution varying wildly with the parameter-value) is surprising, even when the equations are perfectly “decidable”.

scholarly journals Optimal sup norm bounds for newforms on GL2 with maximally ramified central character

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Félicien Comtat
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Modular Forms ◽  
Upper Bound ◽  
Upper Bounds ◽  
Central Character ◽  
Ramanujan Conjecture ◽  
The Moment ◽  
Level Aspect

AbstractRecently, the problem of bounding the sup norms of {L^{2}}-normalized cuspidal automorphic newforms ϕ on {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to\|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon},at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

On some weighted zero-sum constants II

2018 ◽  
Vol 14 (02) ◽  
pp. 383-397 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structure Theorem ◽  
Finite Abelian Group ◽  
Zero Sum

For a finite abelian group [Formula: see text] with exponent [Formula: see text], let [Formula: see text]. The constant [Formula: see text] (respectively [Formula: see text]) is defined to be the least positive integer [Formula: see text] such that given any sequence [Formula: see text] over [Formula: see text] with length [Formula: see text] has a [Formula: see text]-weighted zero-sum subsequence of length [Formula: see text] (respectively at most [Formula: see text]). In [M. N. Chintamani and P. Paul, On some weighted zero-sum constants, Int. J. Number Theory 13(2) (2017) 301–308], we proved the exact value of this constant for the group [Formula: see text] and proved the structure theorem for the extremal sequences related to this constant. In this paper, we prove the similar results for the group [Formula: see text] and we obtained an upper bound when [Formula: see text] is replaced by any integer [Formula: see text].

Mathematical analysis in examples and tasks. Part 2

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Field Theory ◽  
Graduate Students ◽  
Mathematical Analysis ◽  
Differential Calculus ◽  
Advanced Mathematics ◽  
Analytic Geometry ◽  
Several Variables ◽  
Background Material ◽  
Basic Concepts ◽  
Independent Work

The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 2 of the proposed cycle of workshops on the following topics: analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Fourier series; applications to the analysis and solution of applied problems. These topics are studied in universities, usually in the second semester in the discipline "Mathematical analysis" or the course "Higher mathematics", "Mathematics". For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the acquired knowledge in each section designed exercises and tasks with answers and guidance. It is recommended that teachers, students and graduate students studying advanced mathematics.

Pre-Pólya group in even dihedral extensions of ℚ

2020 ◽  
pp. 2150001
Author(s):  
Abbas Maarefparvar
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Upper Bound ◽  
Dihedral Group ◽  
American Mathematical Society ◽  
Number Fields ◽  
Manuscripta Math ◽  
Galois Number ◽  
Dihedral Extensions ◽  
Special Case

Investigating on Pólya groups [P. J. Cahen and J. L. Chabert Integer-Valued Polynomials, Mathematical Surveys and Monographs, Vol. 48 (American Mathematical Society, Providence, 1997)] in non-Galois number fields, Chabert [J. L. Chabert and E. Halberstadt, From Pólya fields to Pólya groups (II): Non-Galois number fields, J. Number Theory (2020), https://doi.org/10.1016/j.jnt.2020.06.008 ] introduced the notion of pre-Pólya group [Formula: see text], which is a generalization of the pre-Pólya condition, duo to Zantema [H. Zantema, Integer valued polynomials over a number field, Manuscripta Math. 40 (1982) 155–203]. In this paper, using class field theory, we describe the pre-Pólya group of a [Formula: see text]-field [Formula: see text], for [Formula: see text] an even integer, where [Formula: see text] denotes the dihedral group of order [Formula: see text]. Moreover, for special case [Formula: see text], we improve the Zantema’s upper bound on the maximum ramification in Pólya [Formula: see text]-fields.

On associating significance levels with temporal changes in empirical orthogonal function analysis: a case study for ENSO

2020 ◽  
Author(s):  
Gabor Drotos
Keyword(s):  
Upper Bound ◽  
Null Hypothesis ◽  
Southern Oscillation ◽  
Function Analysis ◽  
Finite Size ◽  
Temporal Changes ◽  
Empirical Orthogonal Functions ◽  
Max Planck ◽  
Orthogonal Functions ◽  
Significance Level

<p>The availability of a large ensemble enables one to evaluate empirical orthogonal functions (EOFs) with respect to the ensemble without relying on temporal variability at all. Variability across the ensemble at any given time is supposed to represent the most relevant probability distribution for climate-related studies, and this distribution is presumably subject to temporal changes in the presence of time-dependent forcing. Such changes may be observable in spatial patterns of ensemble-based EOFs and associated eigenvalues. Unfortunately, estimates of these changes come with a considerable error due to the finite size of the ensemble, so that associating a significance level with the presence of a change (with respect to a null hypothesis about the absence of any change) should be the first step of analyzing the time evolution.</p><p>It turns out, however, that the conditions for the applicability of usual hypothesis tests about stationarity are not satisfied for the above-mentioned quantities. What proves to be feasible is to estimate an upper bound on the significance level for nonstationarity. This means that the true significance level would ideally be lower or equal to what is estimated, which would prevent unjustified confidence in the detection of nonstationarity (i.e., falsely rejecting the null hypothesis could not become more probable than claimed). Most importantly, one would avoid seriously overconfident conclusions about the sign of the change in this way. Notwithstanding, the estimate for the upper bound on the significance level is also affected by the finite number of the ensemble members. It nevertheless becomes more and more precise for increasing ensemble size and may serve as a first guidance for currently available ensemble sizes.</p><p>The details of the estimation are presented in the example of the EOF-based analysis of the El Niño–Southern Oscillation (ENSO) as it appears in the historical and RCP8.5 simulations of the Max Planck Institute Grand Ensemble. A comparison between results including and excluding ensemble members initialized with an incomplete spinup in system components with a long time scale is also given.</p>

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