AMS Chelsea Publishing 1966; 1602 pp; hardcover Volume: 86 Reprint/Revision History: first AMS printing 1999; reprinted 2002 ISBN10: 0821819380 ISBN13: 9780821819388 List Price: US$152 Member Price: US$136.80 Order Code: CHEL/86.H
 Dickson's History is truly a monumental account of the development of one of the oldest and most important areas of mathematics. It is remarkable today to think that such a complete history could even be conceived. That Dickson was able to accomplish such a feat is attested to by the fact that his History has become the standard reference for number theory up to that time. One need only look at later classics, such as Hardy and Wright, where Dickson's History is frequently cited, to see its importance. The book is divided into three volumes by topic. In scope, the coverage is encyclopedic, leaving very little out. It is interesting to see the topics being resuscitated today that are treated in detail in Dickson. The first volume of Dickson's History covers the related topics of divisibility and primality. It begins with a description of the development of our understanding of perfect numbers. Other standard topics, such as Fermat's theorems, primitive roots, counting divisors, the Möbius function, and prime numbers themselves are treated. Dickson, in this thoroughness, also includes less workhorse subjects, such as methods of factoring, divisibility of factorials and properties of the digits of numbers. Concepts, results and citations are numerous. The second volume is a comprehensive treatment of Diophantine analysis. Besides the familiar cases of Diophantine equations, this rubric also covers partitions, representations as a sum of two, three, four or \(n\) squares, Waring's problem in general and Hilbert's solution of it, and perfect squares in arithmetical and geometrical progressions. Of course, many important Diophantine equations, such as Pell's equation, and classes of equations, such as quadratic, cubic and quartic equations, are treated in detail. As usual with Dickson, the account is encyclopedic and the references are numerous. The last volume of Dickson's History is the most modern, covering quadratic and higher forms. The treatment here is more general than in Volume II, which, in a sense, is more concerned with special cases. Indeed, this volume chiefly presents methods of attacking whole classes of problems. Again, Dickson is exhaustive with references and citations. Table of Contents Part 1, Divisibility and Primality  Perfect, multiply perfect, and amicable numbers
 Formulas for the number and sum of divisors, problems of Fermat and Wallis
 Fermat's and Wilson's theorems, generalizations and converses; symmetric functions of \(1, 2, \dots, p1\), modulo \(p\)
 Residue of \((u^{p1}1)/p\) modulo \(p\)
 Euler's \(\phi\)function, generalizations; Farey series
 Periodic decimal fractions; periodic fractions; factors of \(10^{n}\pm 1\)
 Primitive roots, exponents, indices, binomial congruences
 Higher congruences
 Divisibility of factorials and multinomial coefficients
 Sum and number of divisors
 Miscellaneous theorems on divisibility, greatest common divisor, least common multiple
 Criteria for divisibility by a given number
 Factor tables, lists of primes
 Methods of factoring
 Fermat numbers \(F_{n}=2^{2^n}+1\)
 Factors of \(a^{n}\pm b^{n}\)
 Recurring series; Lucas' \(u_{n}, v_{n}\)
 Theory of prime numbers
 Inversion of functions; Möbius' function \(\mu(n)\); numerical integrals and derivatives
 Properties of the digits of numbers
 Author index
 Subject index
Part 2, Diophantine Analysis  Polygonal, pyramidal and figurate numbers
 Linear diophantine equations and congruences
 Partitions
 Rational right triangles
 Triangles, quadrilaterals, and tetrahedra
 Sum of two squares
 Sum of three squares
 Sum of four squares
 Sum of \(n\) squares
 Number of solutions of quadratic congruences in \(n\) unknowns
 Liouville's series of eighteen articles
 Pell equation; \(ax^2 + bx +c\) made a square
 Further single equations of the second degree
 Squares in arithmetical or geometrical progression
 Two or more linear functions made squares
 Two quadratic functions of one or two unknowns made squares
 Systems of two equations of degree two
 Three or more quadratic functions of one or two unknowns made squares
 Systems of three or more equations of degree two in three or more unknowns
 Quadratic form made an \(n\)th power
 Equations of degree three
 Equations of degree four
 Equations of degree \(n\)
 Sets of integers with equal sums of like powers
 Waring's problem and related results
 Fermat's last theorem, \(ax^{r}+ by^{s}= cz^{t}\), and the congruence \(x^n + y^n\equiv z^n\pmod p\)
 Author index
 Subject index
Part 3, Quadratic and Higher Forms  Reduction and equivalence of binary quadratic forms, representation of integers
 Explicit values of \(x, y\) in \(x^2 +\Delta y^2 =g\)
 Composition of binary quadratic forms
 Orders and genera; their composition
 Irregular determinants
 Number of classes of binary quadratic forms with integral coefficients
 Binary quadratic forms whose coefficients are complex integers or integers of a field
 Number of classes of binary quadratic forms with complex integral coefficients
 Ternary quadratic forms
 Quaternary quadratic forms
 Quadratic forms in \(n\) variables
 Binary cubic forms
 Cubic forms in three or more variables
 Forms of degree \(n\geqq 4\)
 Binary Hermitian forms
 Hermitian forms in \(n\) variables and their conjugates
 Bilinear forms, matrices, linear substitutions
 Representation by polynomials modulo \(p\)
 Congruencial theory of forms
 Author index
 Subject index
