Number Theory in the Spirit of Liouville

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

The only book entirely devoted to Joseph Liouville's elementary method in number theory, this gentle introduction explains his method in a clear and straightforward manner, including many applications. This is an extremely valuable resource suitable for advanced undergraduate and beginning graduate students, and researchers in number theory.Joseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville's ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.