1: A Survey of Divisibility2: Primes and Factorization3: Theory of Modular Arithmetic4: A Survey of Modular Arithmetic with Prime Moduli5: Euler's Generalization of Fermat's Theorem6: Primitive Roots and Indices7: Quadratic Residues8: Non-Linear Diophantine Equations

"This book examines the patterns and beauty of positive integers by using elementary methods. It discusses some of the outstanding problems which have not been resolved even after hundreds of years of trying. A challenging problem, even for powerful computers, is factorizing integers and the book highlights some methods that are used to simplify this. We factorize integers of the type and solve the equivalent non - linear Diophantine equation where p is prime. To see if such equations have integer solutions, we use the 'Law of Quadratic Reciprocity' which is one of the most powerful results in number theory. The methods of factorization use a new arithmetic called 'clock arithmetic' which also helps in finding the last few digits of a large number without writing down all the digits. The book applies clock arithmetic to test whether a given number is prime or composite. We conclude by showing one of the great results of mathematics that a prime number which leaves a reminder of one after dividing by four can be written as the sum of two squares. However, a prime number which leaves a reminder of three after dividing by four cannot be written as the sum of two squares. Most of the results in the book are placed in an historical context"--