Checking the Goldbach conjecture up to 4⋅10¹¹

The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools to attack these problems. This book is intended for students who want to learn additive number theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in number theory and real analysis.

Exact order of subsets of asymptotic bases in additive number theory

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plunnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.

The determination of the minimal s such that all large natural numbers n admit a representation asis an interesting problem in the additive theory of numbers and has a considerable literature, For historical comments the reader is referred to the author's paper [2] where the best currently known result is proved. The purpose here is a further improvement.

In this paper, we consider a question of sum-keeping about a multiplicative subsemigroup and its generator subsets in a semiring, and develop some elementary (collapse) process of the sum-keeping retraction through subsets until one minimal generators subset. As an application, we study and analyze several classical problems in additive number theory on the semiring of non-negative integers by this algebraic and combinatory idea, and provide new proofs in more simple and direct way for several classical results in number theory. Some further questions are also presented and discussed.

An improvement and generalization of Bellman-Shapiro’s theorem on a problem in additive number theory

Lehmer if there was anything known about the representation of an integer A in the form h - In, i-X where all the prime factors of each A,-are of a given form.A search of the literature seemed to indicate that various theorems had been conjectured but none actually proved.fFor example, L. Euler stated without proof that every integer of the form 47+2 is a sum of two primes each of the form 47 + 1.Even the weaker statement that every integer of the form 4/+2 is a sum of two integers which have all their prime factors of the form 4/+1 has not yet been proved.In view of the absence of any definite results in the literature it seems worthwhile to point out that some very interesting theorems can be obtained in an elementary way.This is done in Part I of this paper and the results are summarized in Theorems 1, 2, and 3 below.In Part II we use the method of Viggo BrunJ to prove a general theorem and from this we deduce Theorems 4 and 5 below.

A linear equation on k unknowns is called a (k, h)-equation if it is of the form ∑k i=1 aixi = 0, with ai ∈ {−h, . . . , h} and ∑ ai = 0. For a (k, h)-equation E, let rE(n) denote the size of the largest subset of the first n integers with no solution of E (besides certain trivial solutions). Several special cases of this general problem, such as Sidon’s equation and sets without threeterm arithmetic progressions, are some of the most well studied problems in additive number theory. Ruzsa was the first to address the general problem of the influence of certain properties of equations on rE(n). His results suggest, but do not imply, that for every fixed k, all but an O(1/h) fraction of the (k, h)-equations E are such that rE(n) > n1−o(1). In this paper we address the generalized problem of estimating the size of the largest subset of the first n integers with no solution of a set S of (k, h)-equations (again, besides certain trivial solutions). We denote this quantity by rS(n). Our main result is that all but an O(1/h) fraction of the sets of (k, h)-equations S of size k − b √ 2kc + 1, are such that rS(n) > n1−o(1). We also give several additional results relating properties of sets of equations and rS(n).

Some Applications of a Theorem of M. Kneser.

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.

An edge‐coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge‐coloured graph on vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of for this question. Very recently, Janzer–Sudakov and Kim–Lee–Liu–Tran independently removed the term in Tomon's bound, showing a bound of . We prove an upper bound of for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the term, and so, it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non‐abelian groups.

On a girth–free variant of the Bourgain–Gamburd machine

A method of constructing codes capable of correcting an arbitrary number of asymmetrical errors is given. The method is related to a problem in the additive theory of numbers.

Journal of the London Mathematical SocietyVolume s1-25, Issue 2 p. 115-119 Some aspects of Hardy's mathematical work (4) the Additive Theory of Numbers A. E. Ingham, A. E. InghamSearch for more papers by this author A. E. Ingham, A. E. InghamSearch for more papers by this author First published: April 1950 https://doi.org/10.1112/jlms/s1-25.2.115AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-25, Issue2April 1950Pages 115-119 RelatedInformation

Preface. 1. On analytic continuation of multiple L-functions and related zeta-functions S. Akiyama, H. Ishikawa. 2. On the values of certain q-hypergeometric series II M. Amou, et al. 3. The class number one problem for some non-normal sextic CM-fields G. Boutteaux, S. Louboutin. 4. Ternary problems in additive prime number theory J. Brudern, K. Kawada. 5. A generalization of E. Lehmer's congruence and its applications Tianxin Cai. 6. On Chen's theorem Cai Yingchun, Lu Minggao. 7. On a twisted power mean of L(1,chi) S. Egami. 8. On the pair correlation of the zeros of the Riemann zeta function A. Fujii. 9. Discrepancy of some special sequences K. Goto, Y. Ohkubo. 10. Pade approximation to the logarithmic derivative of the Gauss hypergeometric function M. Hata, M. Huttner. 11. The evaluation of the sum over arithmetic progressions for the co-efficients of the Rankin-Selberg series II Y. Ichihara. 12. Substitutions, atomic surfaces, and periodic beta expansions S. Ito, Y. Sano. 13. The largest prime factor of integers in the short interval Chaohua Jia. 14. A general divisor problem in Landau's framework S. Kanemitsu, A. Sankaranarayanan. 15. On inhomogeneous Diophantine approximation and the Borweins' algorithm, II T. Komatsu. 16. Asymptotic expansions of double gamma-functions and related remarks K. Matsumoto. 17. A note on a certain average of L(1/2 + it, chi) L. Murata. 18. On covering equivalence Zhi-Wei Sun.19. Certain words, tilings, their non-periodicity, and substitutions of high dimension J. Tamura. 20. Determination of all Q-rational CM-points in moduli spaces of polarized abelian surfaces A. Umegaki. 21. On families of cubic Thue equations I. Wakabayashi. 22. Two examples of zeta-regularization M. Yoshimoto. 23. A hybrid mean value formula of Dedekind sums and Hurwitz zeta-functions Zhang Wenpeng.