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Although the Lucas sequences were known to earlier investigators such as Lagrange, Legendre and Genocchi, it is because of the enormous number and variety of results involving them, revealed by Édouard Lucas between 1876 and 1880, that they are now named after him. Since Lucas’ early work, much more has been discovered concerning these remarkable mathematical objects, and the objective of this book is to provide a much more thorough discussion of them than is available in existing monographs. In order to do this a large variety of results, currently scattered throughout the literature, are brought together. Various sections are devoted to the intrinsic arithmetic properties of these sequences, primality testing, the Lucasnomials, some associated density problems and Lucas’ problem of finding a suitable generalization of them. Furthermore, their application, not only to primality testing, but also to integer factoring, efficient solution of quadratic and cubic congruences, cryptography and Diophantine equations are briefly discussed. Also, many historical remarks are sprinkled throughout the book, and a biography of Lucas is included as an appendix. Much of the book is not intended to be overly detailed. Rather, the objective is to provide a good, elementary and clear explanation of the subject matter without too much ancillary material. Most chapters, with the exception of the second and the fourth, will address a particular theme, provide enough information for the reader to get a feel for the subject and supply references to more comprehensive results. Most of this work should be accessible to anyone with a basic knowledge of elementary number theory and abstract algebra. The book’s intended audience is number theorists, both professional and amateur, students and enthusiasts.

Number theory. --- Number Theory.

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Alan Baker's 60th birthday in August 1999 offered an ideal opportunity to organize a conference at ETH Zurich with the goal of presenting the state of the art in number theory and geometry. Many of the leaders in the subject were brought together to present an account of research in the last century as well as speculations for possible further research. The papers in this volume cover a broad spectrum of number theory including geometric, algebrao-geometric and analytic aspects. This volume will appeal to number theorists, algebraic geometers, and geometers with a number-theoretic background. However, it will also be valuable for mathematicians (in particular research students) who would like to be informed of the state of number theory at the start of the 21st century and in possible developments for the future.

Number theory

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This is the fourteenth annual volume arising from the Seminaire de Theorie des Nombres de Paris. As with previous volumes the whole spectrum of number theory is discussed, with many contributions from some of the world's leading figures. The very latest research developments are covered and much of the work presented here will not be found elsewhere. Also included are surveys that will serve to guide the reader through the extensive published literature. This will be a necessary addition to the libraries of all workers in number theory.

Number theory

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This volume is based on the successful 6th China-Japan Seminar on number theory that was held in Shanghai Jiao Tong University in August 2011. It is a compilation of survey papers as well as original works by distinguished researchers in their respective fields. The topics range from traditional analytic number theory - additive problems, divisor problems, Diophantine equations - to elliptic curves and automorphic L-functions. It contains new developments in number theory and the topics complement the existing two volumes from the previous seminars which can be found in the same book series.

Number theory

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This handbook covers a wealth of topics from number theory, special attention being given to estimates and inequalities. As a rule, the most important results are presented, together with their refinements, extensions or generalisations. These may be applied to other aspects of number theory, or to a wide range of mathematical disciplines. Cross-references provide new insight into fundamental research. Audience: This is an indispensable reference work for specialists in number theory and other mathematicians who need access to some of these results in their own fields of research.

Number theory.