Elementary Number Theory in Nine ChaptersThis book is intended to serve as a one-semester introductory course in number theory. Throughout the book a historical perspective has been adopted and emphasis is given to some of the subject's applied aspects; in particular the field of cryptography is highlighted. At the heart of the book are the major number theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler, and to fully illustrate the properties of numbers and concepts developed in the text, a wealth of exercises have been included. It is assumed that the reader will have 'pencil in hand' and ready access to a calculator or computer. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject. |
Contents
Preface | 7 |
The intriguing natural numbers | 18 |
Divisibility | 49 |
Prime numbers | 79 |
Perfect and amicable numbers | 127 |
Modular arithmetic | 150 |
Congruences of higher degree | 182 |
Cryptology | 210 |
Representations | 239 |
Partitions | 284 |
Tables | 305 |
The values of tn on pn µn wn | 312 |
| 390 | |
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Common terms and phrases
a₁ arithmetic Arithmetica called canonical representation cipher ciphertext congruences conjecture consecutive continued fraction contradiction convergents coprime cubes Decipher denote the number Determine digital root divides divisible element enciphered equal equation established Euclidean algorithm Euler example Exercises exist integers Fermat Fermat's Little Theorem Ferrers diagram finite form 4k formula Gauss gcd(a gcd(m given greatest common divisor Hence implies induction infinite number integral squares letters mathematical mathematical induction mathematician Mersenne method modulo multiplicative natural numbers number of distinct number of partitions number theoretic function number theory obtain odd number odd prime perfect number plaintext polynomial positive integer Poulet prime factors prime numbers primitive root problem Proof Prove Pythagorean triple quadratic residue rational number residue system modulo result follows sequence Show solve square number superincreasing sequence Suppose Table term Theorem triangle triangular numbers