History of the Theory of Numbers, Volume I Divisibility and Primality

by Dickson, Leonard Eugene

• ISBN13: 9780486789163
• ISBN10: 0486789160
• Format: Hardcover
• Copyright: 2013-12-23
• Publisher: Dover Publications

Summary

The three-volume series History of the Theory of Numbers is the work of the distinguished mathematician Leonard Eugene Dickson, who taught at the University of Chicago for four decades and is celebrated for his many contributions to number theory and group theory. This first volume in the series, which is suitable for upper-level undergraduates and graduate students, is devoted to the subjects of divisibility and primality. It can be read independently of the succeeding volumes, which explore diophantine analysis and quadratic and higher forms.
Within the twenty-chapter treatment are considerations of perfect, multiply perfect, and amicable numbers; formulas for the number and sum of divisors and problems of Fermat and Wallis; Farey series; periodic decimal fractions; primitive roots, exponents, indices, and binomial congruences; higher congruences; divisibility of factorials and multinomial coefficients; sum and number of divisors; theorems on divisibility, greatest common divisor, and least common multiple; criteria for divisibility by a given number; factor tables and lists of primes; methods of factoring; Fermat numbers; recurring series; the theory of prime numbers; inversion of functions; properties of the digits of numbers; and many other related topics. Indexes of authors cited and subjects appear at the end of the book.

Table of Contents

I. Perfect, multiply perfect, and amicable numbers
II. Formulas for the number and sum of divisors, problems of Fermat and Wallis
III. Fermat’s and Wilson’s theorems, generalizations and converses; symmetric functions of 1, 2, ..., p-1, modulo p
IV Residue of (up-1-1)/p modulo p
V. Euler’s function, generalizations; Farey series
VI. Periodic decimal fractions; periodic fractions; factors of 10n
VII. Primitive roots, exponents, indices, binomial congruences
VIII. Higher congruences
IX. Divisibility of factorials and multinomial coefficients
X. Sum and number of divisors
XI. Miscellaneous theorems on divisibility, greatest common divisor, least common multiple
XII. Criteria for divisibility by a given number
XIII. Factor tables, lists of primes
XIV. Methods of factoring
XV. Fermat numbers
XVI. Factors of an+bn
XVII. Recurring series; Lucas’ un, vn
XVIII. Theory of prime numbers
XIX. Inversion of functions; Möbius’ function; numerical integrals and derivatives
XX. Properties of the digits of numbers
Indexes