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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.Â
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.
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