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Full Description
An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history. Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers. Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition. Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject. In this updated edition, you will find a new chapter which brings the reader from undergraduate calculus into analytic number theory. A new section adds a second proof of quadratic reciprocity due to Gauss and Eisenstein. Hundreds of minor edits correct and improve the original edition.
Contents
Seeing arithmetic; Foundations; The Euclidean algorithm; Prime factorization; Rational and constructible numbers; Gaussian and Eisenstein integers; Analytic methods; Modular arithmetic; The modular worlds; Modular dynamics; Assembling the modular worlds; Quadratic residues; Quadratic forms; The topograph; Definite forms; Indefinite forms; Index of theorems; Index of terms; Index of names; Bibliography
