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"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"--

Number theory. --- Number study --- Numbers, Theory of --- Algebra --- Liouville, Joseph, --- Liouville, Joseph

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The theory of numbers continues to occupy a central place in modern mathematics because of both its long history over many centuries as well as its many diverse applications to other fields such as discrete mathematics, cryptography, and coding theory. The proof by Andrew Wiles (with Richard Taylor) of Fermat’s last theorem published in 1995 illustrates the high level of difficulty of problems encountered in number-theoretic research as well as the usefulness of the new ideas arising from its proof. The thirteenth conference of the Canadian Number Theory Association was held at Carleton University, Ottawa, Ontario, Canada from June 16 to 20, 2014. Ninety-nine talks were presented at the conference on the theme of advances in the theory of numbers. Topics of the talks reflected the diversity of current trends and activities in modern number theory. These topics included modular forms, hypergeometric functions, elliptic curves, distribution of prime numbers, diophantine equations, L-functions, Diophantine approximation, and many more. This volume contains some of the papers presented at the conference. All papers were refereed. The high quality of the articles and their contribution to current research directions make this volume a must for any mathematics library and is particularly relevant to researchers and graduate students with an interest in number theory. The editors hope that this volume will serve as both a resource and an inspiration to future generations of researchers in the theory of numbers.

Number theory --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Mathematics. --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of