Number Theory

Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as...

This basic introduction to number theory is ideal for those with no previous knowledge of the subject. The main topics of divisibility, congruences, and the distribution of prime numbers are covered. Of particular interest is the inclusion of a proof for...

This book discusses the p-adic modular forms, the eigencurve that parameterize them, and the p-adic L-functions one can associate to them. These theories and their generalizations to automorphic forms for group of higher ranks are of fundamental...

A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a...

The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed...

This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. My purpose is to make...

Volume I is the first part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid,...

This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums. These arise in many problems in analytic number theory. It is the first cohesive account of much of this material and...

The amount of mathematics invented for number-theoretic reasons is impressive. It includes much of complex analysis, the re-foundation of algebraic geometry on commutative algebra, group cohomology, homological algebra, and the theory of motives. Zeta...

Originally published in 1934 in the Cambridge Tracts, this volume presents the theory of the distribution of the prime numbers in the series of natural numbers. The major part of the book is devoted to the analytical theory founded on the zeta-function...

The study of geometric discrepancy, which provides a framework for quantifying the quality of a distribution of a finite set of points, has experienced significant growth in recent decades. This book provides a self-contained course in number theory,...

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This book provides a clear and accessible introduction to modern algebraic number theory, with a special emphasis on class field theory. Drawing from courses and lectures delivered in seven countries, the text balances simplicity with depth, making...

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Random numbers are immensely important in scientific research, in economic decision making, in polling, gaming, and cryptography and algorithm design. Surprisingly, while many popular books in mathematics have been written about prime numbers, about ,...

Why are there more poor people with small bank accounts than rich people with big bank accounts? Why are there more small planets and stars than big ones in the cosmos? And why are there more small rivers than big rivers, and more harmless tremors than...

Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their...

Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their...

The book explores and investigates a long-standing mathematical question whether a product of two or more positive integers in an arithmetic progression can be a square or a higher power. It investigates, more broadly, if a product of two or more...

Fractions are everywhere and yet most of us learn only basic and rather dry facts about fractions in primary school. This book makes fractions come to life in a friendly, lively, and accessible way, detailing the history of fractions and their crucial...

This book serves as an illuminating introduction to the intricacies of the geometry of numbers. It commences by exploring basic concepts of convex sets and lattices in Euclidean space and goes on to delve into Minkowski’s fundamental theorem for convex...

This book proposes a novel approach to the study of Diophantine equations: define an appropriate version of the equation’s size, order all polynomial Diophantine equations by size, and then solve the equations in order. Natural questions about the...