Quadratic Diophantine equations. With a foreword by Preda Mihăilescu (Q935684)

The book under review is an excellent book on the interesting subject of quadratic Diophantine equations. It is well written, well organized, and contains a wealth of material that one does not expect to find in a book of its size, with full proofs of scores of theorems. It also has a long list of references containing 232 items. This reviewer does not know any book that covers similar material, and sees it as a very valuable and much needed addition to the literature on number theory. The book is written masterly, with much of the material based on the authors' own contributions to the subject. Most of these contributions had already appeared in previous papers, but some are new. The book has 7 chapters. Chapter 1 describes 10 contexts in which quadratic Diophantine equations appear. These include Hilbert's tenth problem, Euler's concordant forms, ranks of Mordell-Weil groups, Hecke groups, etc. Chapter 2 deals with continued fractions and Diophantine approximation, and introduces quadratic rings. Chapter 3 gives 3 different ways for finding the general solution of Pell's equation \(x^2 - Dy^2 = 1\), namely, elementary methods, continued fractions, and quadratic rings. It also contains treatments of the more general equation \(ax^2 - by^2 = 1\) and the negative Pell's equation \(x^2 - Dy^2 = -1\). Chapter 4 is concerned with the general Pell's equation \(x^2 - Dy^2 = N\). It contains tests for the solvability of this equation, and different algorithms for solving it and solving the general negative Pell's equation. Among other things, it treats the special equation \(x^2 - Dy^2 = \pm 4\) and the equations \(x^2 - Dy^2 = N\) for \(N < \sqrt{D}\), and gives numerical examples. It also treats the equations \(ax^2 - by^2 = c\) and \(ax^2 + bxy +cy^2 = N\). The last section treats the special cases \(x^2 - Dy^2 = \pm N\) for \(2 \leq D \leq 7\). Chapter 5 consists of treatments of 16 interesting types of equations that can be reduced to Pell-like equations. Chapter 6 is about Diophantine representations of sequences. It focuses on Fibonacci, Lucas, and generalized Lucas sequences, and establishes improvements of some existing results. Chapter 7 contains several applications. These include the problem of when both \(an+b\) and \(cn+d\) are perfect squares, when a rational number is a quotient of triangular numbers, when the quotient of 2 triangular numbers is a perfect square, and whether an \(n\)-gonal number can be written as a product of 2 \(n\)-gonal numbers. It also answers questions about powerful numbers, and about certain integral \((2 \times 2)\)-matrices.