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2009 | Buch

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Number Theory

An Introduction to Mathematics

verfasst von: W.A. Coppel

Verlag: Springer New York

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included.

The book is divided into two parts. Part A covers key concepts of number theory and could serve as a first course on the subject. Part B delves into more advanced topics and an exploration of related mathematics. Part B contains, for example, complete proofs of the Hasse-Minkowski theorem and the prime number theorem, as well as self-contained accounts of the character theory of finite groups and the theory of elliptic functions. The prerequisites for this self-contained text are elements from linear algebra. Valuable references for the reader are collected at the end of each chapter. It is suitable as an introduction to higher level mathematics for undergraduates, or for self-study.

Inhaltsverzeichnis

Frontmatter

I. The Expanding Universe of Numbers

Abstract

For many people, numbers must seem to be the essence of mathematics. Number theory, which is the subject of this book, is primarily concerned with the properties of one particular type of number, the ‘whole numbers’ or integers. However, there are many other types, such as complex numbers and p-adic numbers. Somewhat surprisingly, a knowledge of these other types turns out to be necessary for any deeper understanding of the integers.

In this introductory chapter we describe several such types (but defer the study of p-adic numbers to Chapter VI). To embark on number theory proper the reader may proceed to Chapter II now and refer back to the present chapter, via the Index, only as occasion demands.

W. A. Coppel

II. Divisibility

Abstract

In the set ℕ of all positive integers we can perform two basic operations: addition and multiplication. In this chapter we will be primarily concerned with the second operation.

W. A. Coppel

III. More on Divisibility

Abstract

In this chapter the theory of divisibility is developed further. The various sections of the chapter are to a large extent independent.We consider in turn the law of quadratic reciprocity, quadratic fields, multiplicative functions, and linear Diophantine equations.

W. A. Coppel

IV. Continued Fractions and Their Uses

Abstract

Let $\xi = \xi_0$ be an irrational real number. Then we can write

$$\xi_0 = {\rm a}_0 + \xi^{-1}_{1},$$

where $a_0 = \lfloor \xi_0 \rfloor$ is the greatest integer $\leq \xi_0$ and where $\xi_1 > 1$ is again an irrational number. Hence the process can be repeated indefinitely:

$$\begin{array}{c} \xi_1 = {\rm a}_1 + \xi^{-1}_2, \quad ({\rm a}_1 = \lfloor \xi_1 \rfloor, \xi_2 > 1),\\ \xi_2 = {\rm a}_2 + \xi^{-1}_3, \quad ({\rm a}_2 = \lfloor \xi_2 \rfloor, \xi_3 > 1),\\ \ldots \end{array}$$

By construction, ${\rm a}_n \in \mathbb{Z}$ for all $n \geq 0$ and $a_n \geq 1 \,{\rm if}\, n \geq 1$. The uniquely determined infinite sequence $[{\rm a}_0, {\rm a}_1, {\rm a}_2, \ldots]$ is called the continued fraction expansion of $\xi$. The continued fraction expansion of $\xi_n {\rm is} [{\rm a}_n, {\rm a}_{n+1}, {\rm a}_{n+2}, \ldots]$.

W. A. Coppel

V. Hadamard’s Determinant Problem

Abstract

It was shown by Hadamard (1893) that, if all elements of an n × n matrix of complex numbers have absolute value at most μ, then the determinant of the matrix has absolute value at most μ ⁿ n ^n/2. For each positive integer n there exist complex n × n matrices for which this upper bound is attained. For example, the upper bound is attained for μ = 1 by the matrix (ω ^jk)(1 ≤ j, k ≤ n), where ω is a primitive n-th root of unity. This matrix is real for n = 1, 2. However, Hadamard also showed that if the upper bound is attained for a real n × n matrix, where n > 2, then n is divisible by 4.

W. A. Coppel

VI. Hensel’s p-adic Numbers

Abstract

Hensel simply defined p-adic integers by their power series expansions. We will adopt a more general approach, due to Kürschák (1913), which is based on absolute values.

W. A. Coppel

VII. The Arithmetic of Quadratic Forms

Abstract

We have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask which integers can be represented in the form x ² + 2y ² or, more generally, in the form ax ² + 2bxy + cy ², where a, b, c are given integers. The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange’s theorem that every positive integer is a sum of four squares, was continued during the 19th century by Dirichlet, Hermite, H.J.S. Smith, Minkowski and others. In the 20th century Hasse and Siegel made notable contributions. With Hasse’s work especially it became apparent that the theory is more perspicuous if one allows the variables to be rational numbers, rather than integers. This opened the way to the study of quadratic forms over arbitrary fields, with pioneering contributions by Witt (1937) and Pfister (1965–67).

W. A. Coppel

VIII. The Geometry of Numbers

Abstract

Minkowski (1891) found a new and more geometric proof of Hermite’s result, which gave a much smaller value for the constant c _n. Soon afterwards (1893) he noticed that his proof was valid not only for an n-dimensional ellipsoid f (x) ≤ const., but for any convex body which was symmetric about the origin. This led him to a large body of results, to which he gave the somewhat paradoxical name ‘geometry of numbers’. It seems fair to say that Minkowski was the first to realize the importance of convexity for mathematics, and it was in his lattice point theorem that he first encountered it.

W. A. Coppel

IX. The Number of Prime Numbers

Abstract

It was already shown in Euclid’s Elements (Book IX, Proposition 20) that there are infinitely many prime numbers. The proof is a model of simplicity: let $p_1, \ldots, p_n$ be any finite set of primes and consider the integer $N = p_1 \ldots p_n + 1$. Then $N > 1$ and each prime divisor p of N is distinct from $p_1, \ldots, p_n$, since $p = p_j$ would imply that p divides $N - p_1 \cdots p_n = 1$. It is worth noting that the same argument applies if we take $N = p^{\propto_1}_1 \cdots p^{\propto_n}_n + 1$, with any positive integers $\propto_1, \ldots, \propto_n$.

W. A. Coppel

X. A Character Study

Abstract

Let a and m be integers with $1 \leq a < m$. If a and m have a common divisor d > 1, then no term after the first of the arithmetic progression

$$a, a + m, a + 2m,\ldots$$

(*)

is a prime. Legendre (1788) conjectured, and later (1808) attempted a proof, that if a and m are relatively prime, then the arithmetic progression (*) contains infinitely many primes.

W. A. Coppel

XI. Uniform Distribution and Ergodic Theory

Abstract

A trajectory of a system which is evolving with time may be said to be ‘recurrent’ if it keeps returning to any neighbourhood, however small, of its initial point, and ‘dense’ if it passes arbitrarily near to every point. It may be said to be ‘uniformly distributed’ if the proportion of time it spends in any region tends asymptotically to the ratio of the volume of that region to the volume of the whole space. In the present chapter these notions will be made precise and some fundamental properties derived. The subject of dynamical systems has its roots in mechanics, but we will be particularly concerned with its applications in number theory.

W. A. Coppel

XII. Elliptic Functions

Abstract

Our discussion of elliptic functions may be regarded as an essay in revisionism, since we do not use Liouville’s theorem, Riemann surfaces or the Weierstrassian functions. We wish to show that the methods used by the founding fathers of the subject provide a natural and rigorous approach, which is very well suited for applications.

The work is arranged so that the initial sections aremutually independent, although motivation for each section is provided by those which precede it. To some extent we have also separated the discussion for real and for complex parameters, so that those interested only in the real case may skip the complex one.

W. A. Coppel

XIII. Connections with Number Theory

Abstract

In Proposition II.40 we proved Lagrange’s theorem that every positive integer can be represented as a sum of 4 squares. Jacobi (1829), at the end of his Fundamenta Nova, gave a completely different proof of this theorem with the aid of theta functions. Moreover, his proof provided a formula for the number of different representations. Hurwitz (1896), by developing further the arithmetic of quaternions which was used in Chapter II, also derived this formula. Here we give Jacobi’s argument preference since, although it is less elementary, it is more powerful.

W. A. Coppel

Backmatter

Titel: Number Theory
verfasst von: W.A. Coppel
Verlag: Springer New York
Electronic ISBN: 978-0-387-89486-7
Print ISBN: 978-0-387-89485-0
DOI: https://doi.org/10.1007/978-0-387-89486-7