# The Theory of Math

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## Overview

The purpose of this little book is to give the reader a convenient introduction to the theory of numbers, one of the most extensive and most elegant disciplines in the whole body of mathematics. The arrangement of the material is as follows:

The first five chapters are devoted to the development of those elements which are essential to any study of the subject. The sixth and last chapter is intended to give the reader some indication of the direction of further study with a brief account of the nature of the material in each of the topics suggested. The treatment throughout is made as brief as is possible consistent with clearness and is confined entirely to fundamental matters. This is done because it is believed that in this way the book may best be made to serve its purpose as an introduction to the theory of numbers.

Numerous problems are supplied throughout the text. These have been selected with great care so as to serve as excellent exercises for the student's introductory training in the methods of number theory and to afford at the same time a further collection of useful results. The exercises marked with a star are more difficult than the others; they will doubtless appeal to the best students.

Finally, I should add that this book is made up from the material used by me in lectures in Indiana University during the past two years; and the selection of matter, especially of exercises, has been based on the experience gained in this way.

R. D. Carmichael.

Contents:
1 ELEMENTARY PROPERTIES OF INTEGERS
1 Fundamental Notions and Laws
2 Definition of Divisibility The Unit
3 Prime Numbers The Sieve of Eratosthenes
4 The Number of Primes is Infinite
5 The Fundamental Theorem of Euclid
6 Divisibility by a Prime Number
7 The Unique Factorization Theorem
8 The Divisors of an Integer
9 The Greatest Common Factor of Two or More Integers
10 The Least Common Multiple of Two or More Integers
11 Scales of Notation
12 Highest Power of a Prime p Contained in n!
13 Remarks Concerning Prime Numbers
2 ON THE INDICATOR OF AN INTEGER
1 Definition
2 The Indicator of a Product
3 The Indicator of any Positive Integer
4 Sum of the Indicators of the Divisors of a Number
3 ELEMENTARY PROPERTIES OF CONGRUENCES
1 Congruences Modulo m
2 Solutions of Congruences by Trial
3 Properties of Congruences Relative to Division
4 Congruences with a Prime Modulus
5 Linear Congruences
4 THE THEOREMS OF FERMAT AND WILSON
1 Fermat's General Theorem
2 Euler's Proof of the Simple Fermat Theorem
3 Wilson's Theorem
4 The Converse of Wilson's Theorem
5 Impossibility of 1 . 2 . 3... n . 1 + 1 = n^k for n > 5
6 Extension of Fermat's Theorem
7 On the Converse of Fermat's Simple Theorem
8 Application of Previous Results to Linear Congruences
9 Application of the Preceding Results to the Theory of Quadratic Residues
5 PRIMITIVE ROOTS MODULO m
1 Exponent of an Integer Modulo m
2 Another Proof of Fermat's General Theorem
3 Denition of Primitive Roots
4 Primitive roots modulo p
5 Primitive Roots Modulo p, p an Odd Prime
6 Primitive Roots Modulo 2p, p an Odd Prime
7 Recapitulation
8 Primitive -roots
6 OTHER TOPICS
1 Introduction