"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"--

An introductory textbook with a unique historical approach to teaching number theory The natural numbers have been studied for thousands of years, yet most undergraduate textbooks present number theory as a long list of theorems with little mention of how these results were discovered or why they are important. This book emphasizes the historical development of number theory, describing methods, theorems, and proofs in the contexts in which they originated, and providing an accessible introduction to one of the most fascinating subjects in mathematics. Written in an informal style by an award-winning teacher, Number Theory covers prime numbers, Fibonacci numbers, and a host of other essential topics in number theory, while also telling the stories of the great mathematicians behind these developments, including Euclid, Carl Friedrich Gauss, and Sophie Germain. This one-of-a-kind introductory textbook features an extensive set of problems that enable students to actively reinforce and extend their understanding of the material, as well as fully worked solutions for many of these problems. It also includes helpful hints for when students are unsure of how to get started on a given problem. Uses a unique historical approach to teaching number theory Features numerous problems, helpful hints, and fully worked solutions Discusses fun topics like Pythagorean tuning in music, Sudoku puzzles, and arithmetic progressions of primes Includes an introduction to Sage, an easy-to-learn yet powerful open-source mathematics software package Ideal for undergraduate mathematics majors as well as non-math majors Digital solutions manual (available only to professors)

This book is a first course in statistics for students of biology. Most of the examples have an ecological bias, but illustrate principles which have direct relevance for biologists doing laboratory work. The structured approach begins with basic concepts, and progresses towards an appreciation of the needs and use of analysis of variance and regression, and includes the use of computer statistical packages. The work is clearly explained with worked examples of real-life biological problems, and should be suitable for undergraduate students engaged in quantitative biological work. Biostatistics should give students a sound grasp of the key principles of biological statistics without overwhelming detail, and should allow students to quickly apply techniques to their own work and data.

A gentle introduction to Liouville's powerful method in elementary number theory. Suitable for advanced undergraduate and beginning graduate students.