Number Theory with Computations
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Beschreibung
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 Tonelli–Shanks algorithm, factorisation methods, continued fraction representations of e, and the irrationality of ���(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.
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 Tonelli–Shanks 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
Autor*in
Peter Shiu
Themen in »Number Theory with Computations«
undergraduate textbook in analytic number theory Number theory with Python circle method Continued fractions Euclidean algorithm Meissel-Lehmer method sieve of Erastothenes Gauss sum Pell’s equation Dirichlet characters Waring’s problem Vinogradov’s theorem highly composite numbers smooth numbers sparsely totient numbers