Number Theory with Computations

This introductory text is designed for undergraduate courses in number theory, covering both elementary number theory and analytic number theory. The book emphasises computational aspects, including algorithms and their implementation in Python.

The book is divided into two parts. The first part, on elementary number theory, deals with concepts such as induction, divisibility, congruences, primitive roots, cryptography, and continued fractions. The second part is devoted to analytic number theory and includes chapters on Dirichlet's theorem on primes in arithmetic progressions, the prime number theorem, smooth numbers, and the famous circle method of Hardy and Littlewood. The book contains many topics not often found in introductory textbooks, such as Aubry's theorem, the TonelliShanks algorithm, factorisation methods, continued fraction representations of e , and the irrationality of 𝜁(3). Each chapter concludes with a summary and notes, as well as numerous exercises.

Assuming only basic calculus for the first part of the book, the second part assumes some knowledge of complex analysis. Familiarity with basic coding syntax will be helpful for the computational exercises.

Provides a unified introduction to elementary and analytic number theory Includes algorithms in Python for a computational approach to number theory Contains over 200 exercises with hints

Autorentext

Peter Shiu, now retired, was a Reader in pure mathematics at the University of Loughborough. The author of over 30 research papers, and some 50 expository articles, mainly in number theory, he served as the United Kingdom Team Leader at the 31st International Mathematical Olympiad (1990) in Beijing, China. Peter also translated works of the distinguished Chinese mathematician Hua Loo-Keng, and he is currently a reviewer in number theory for Mathematical Reviews.

Inhalt

Part I Elementary Number Theory.- 1 Basics.- 2 Arithmetic functions I.- 3 Prime numbers: Euclid and Eratosthenes.- 4 Quadratic residues and congruences.- 5 Primitive roots.- 6 Sums of squares.- 7 Continued fractions.- Part II Analytic Number Theory.- 8 Diophantine approximations.- 9 Distribution of prime numbers.- 10 Arithmetic functions II.- 11 Prime number theorem.- 12 Primes in arithmetic progressions.- 13 Smooth numbers.- 14 Circle method.