To the reader

This textbook covers with the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense, but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. Nine main areas of the subject are treated: (i) divisibility and multiplicative functions; (ii) congruences and quadratic residues; (iii) the basics of algebraic numbers and sums of squares; (iv) continued fractions, diophantine approximations and transcendence; (v) quadratic forms; (vi) partitions; (vii) the prime numbers; (viii) diophantine equations; and (ix) elliptic curves (this part has been considerably extended for this edition). The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on fermat's last theorem is also breifly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables.

This chapter is our second digression into number theory. We saw in §1.8 that Jensen’s Formula (or the First Main Theorem) can be viewed as an analogue of the Artin-Whaples product formula in number theory. In the present chapter, we discuss a celebrated number theory theorem known as Roth’s Theorem, and we explain how this is analogous to a weak form of the Second Main Theorem. In fact, it was the formal similarity between Nevanlinna’s Second Main Theorem and Roth’s Theorem that led C. F. Osgood (see [Osg 1981] and [Osg 1985]) to the discovery of an analogy between Nevanlinna theory and Diophantine approximations. As we said previously, such an analogy was later, but independently, explored in greater depth by P. Vojta. In this chapter we will discuss, without proof, some of the key points in Vojta’s monograph [Vojt 1987], and we include Vojta’s so called “dictionary” relating Nevanlinna theory and number theory. This section is intended for the analytically inclined and is only intended to provide the most basic insight into the beautiful analogy between Nevanlinna theory and Diophantine approximation theory. By omitting proofs, we have tried to make this section less demanding on the reader than [Vojt 19871. However, a true appreciation for Vojta’s analogy cannot be obtained without also studying the proofs of the number theoretic analogues in their full generality. Any reader that is seriously interested in the connection between Nevanlinna theory and Diophantine approximation is highly encouraged to carefully read Vojta’s monograph [Vojt 1987].

This volume contains selected contributions from a very successful meeting on Number Theory and Dynamical Systems held at the University of York in 1987. There are close and surprising connections between number theory and dynamical systems. One emerged last century from the study of the stability of the solar system where problems of small divisors associated with the near resonance of planetary frequencies arose. Previously the question of the stability of the solar system was answered in more general terms by the celebrated KAM theorem, in which the relationship between near resonance (and so Diophantine approximation) and stability is of central importance. Other examples of the connections involve the work of Szemeredi and Furstenberg, and Sprindzuk. As well as containing results on the relationship between number theory and dynamical systems, the book also includes some more speculative and exploratory work which should stimulate interest in different approaches to old problems.

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.

Contributors

For the past 25 years, the Geometrization Program of Thurston has been a driving force for research in 3-manifold topology. This has inspired a surge of activity investigating hyperbolic 3-manifolds (and Kleinian groups), as these manifolds form the largest and least well-understood class of compact 3-manifolds. Familiar and new tools from diverse areas of mathematics have been utilized in these investigations, from topology, geometry, analysis, group theory, and from the point of view of this book, algebra and number theory. This book is aimed at readers already familiar with the basics of hyperbolic 3-manifolds or Kleinian groups, and it is intended to introduce them to the interesting connections with number theory and the tools that will be required to pursue them. While there are a number of texts which cover the topological, geometric and analytical aspects of hyperbolic 3-manifolds, this book is unique in that it deals exclusively with the arithmetic aspects, which are not covered in other texts. Colin Maclachlan is a Reader in the Department of Mathematical Sciences at the University of Aberdeen in Scotland where he has served since 1968. He is a former President of the Edinburgh Mathematical Society. Alan Reid is a Professor in the Department of Mathematics at The University of Texas at Austin. He is a former Royal Society University Research Fellow, Alfred P. Sloan Fellow and winner of the Sir Edmund Whittaker Prize from The Edinburgh Mathematical Society. Both authors have published extensively in the general area of discrete groups, hyperbolic manifolds and low-dimensional topology.

Publisher Summary This chapter discusses the problems and results on combinatorial number theory. It discusses number theoretic problems that are of combinatorial nature. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers along with the properties of objects made out of integers. Integers can be considered either in themselves or as solutions to equations. Questions in number theory are often understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. Real numbers can also be studied in relation to rational numbers.

In recent years, due to work of Vojta, Lang, Osgood, etc., people have started to realize that there is a close relationship between Nevanlinna theory and Diophantine approximation. P. Vojta has compiled a dictionary (see Vojta [V1]) providing correspondences for the concepts fundamental to the two theories. The second main theorem for meromorphic functions is the most important result in Nevanlinna theory, and its counterpart in number theory, via Vojta’s dictionary, is Roth’s theorem. In 1929, R. Nevanlinna conjectured that the second main theorem remains correct if distinct points in P1(C) are replaced by distinct holomorphic maps ai : C → P1(C) with Tai (r) = o(Tf (r)), where Tf (r) is the Nevanlinna characteristic function of the holomorphic map f : C → P1(C) (see [RS 1] for the definition). After many attempts by various people, Osgood proved the conjecture in 1985, which is now known as the Second Main Theorem with Moving Targets. In 1986, Steinmetz gave another simple and elegant proof. Recently, Vojta ([V 2] and [V 3]) formulated and proved its counterpart in number theory: Theorem A (Roth’s theorem with moving targets). Let k be a number field and let S be a finite set of places of k . Let q ∈ Z>0, and let > 0. Then there does not exist an infinite sequence of distinct tuples

The geometry. of numbers can be traced back at least to Lagrange [1773], who proved important results about quadratic forms in two variables. The proofs as well as the formulations of results were purely arithmetic. Reviewing a book of Seeber [1831B] on ternary quadratic forms, Gauß [1831] introduced for the first time geometric methods. Geometric methods were predominant in the work of Dirichlet [1850]. On the other hand Hermite [1850] and Korkine and Zolotareff [1872], [1873], [1877] gave arithmetic proofs for their results on quadratic forms in more than three variables. Finally Minkowski [1891] noticed that a simple geometric argument which he used to give a new proof of a theorem of Hermite could be adapted to much more general situations. Then Minkowski [1896B], [7B], [11B] started a systematic study of geometric methods in number theory and called this new branch of number theory geometry of numbers. Many results and most concepts of modern geometry of numbers have their origin in the work of Minkowski. After Minkowski many eminent mathematicians made contributions to this field. In order to avoid controversies I will not mention any of them. Geometry of numbers is closely related to other branches of number theory such as algebraic number theory and Diophantine approximation. A flourishing offspring is discrete geometry, developed mainly by Fejes Tóth and his school.

Summary This paper formalizes problems 38, 58, 160, 164, 168, 171, 188, 195, 196, and 198 from “250 Problems in Elementary Number Theory” by Wacław Sierpiński.

Contributors

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.

Through basic research in mathematics, the NTT Institute for Fundamental Mathematics aims to enrich the "fountain of knowledge" that nourishes science and technology.In this article, we first provide an overview of the research being carried out at the Institute then introduce the Institute's core research areas: number theory, especially arithmetic dynamics; algebraic and arithmetic geometry; and representation theory and automorphic forms.

Artūras Dubickas received his Ph.D. from the Moscow State University in 1990. He then had a postdoctoral position at the University of Witwatersrand in Johannesburg. Since 1994 he has a position at Vilnius University (Lithuania), where he is now chief research fellow and professor in the Department of Mathematics and Informatics. His main fields of research are algebraic number theory and transcendence theory.

Methods of the theory of transcendental numbers, Diophantine approximations and solutions of Diophantine equations