A deep understanding of prime numbers is one of the great challenges in mathematics. In this new edition, fundamental theorems, challenging open problems, and the most recent computational records are presented in a language without secrets. The impressive wealth of material and references will make this book a favorite companion and a source of inspiration to all readers.

Paulo Ribenboim is Professor Emeritus at Queen's University in Canada, Fellow of the Royal Society of Canada, and recipient of the George Polya Award of the Mathematical Association of America. He is the author of...

Fermat's problem, also ealled Fermat's last theorem, has attraeted the attention of mathematieians far more than three eenturies. Many clever methods have been devised to attaek the problem, and many beautiful theories have been ereated with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topie is presented in the form of leetures, where I survey the main lines of work on the problem. In the first two leetures, there is a very brief deseription of the early history, as well as a seleetion of a few of the more representative reeent results....

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium senes. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book of Records, reminded me very gently that the most "innumerate" people of the world are of a certain trible in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes, Morris, I'm from Brazil, but my book will contain numbers different from .one.''' He added that the most boring 800-page book is by...

This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples. The introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields. Part one is devoted to residue classes and quadratic residues. In part two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, inertia and ramification of ideals. part three is devoted to Kummers theory of cyclotomic fields, and includes...

ItisnowwellknownthatFermat slasttheoremhasbeenproved. For more than three and a half centuries, mathematicians from the greatnamestothecleveramateurs triedtoproveFermat sfamous statement. The approach was new and involved very sophisticated theories. Finallythelong-soughtproofwasachieved. Thearithmetic theory of elliptic curves, modular forms, Galois representations, and their deformations, developed by many mathematicians, were the tools required to complete the di?cult proof. Linked with this great mathematical feat are the names of TANI- YAMA, SHIMURA, FREY, SERRE, RIBET, WILES, TAYLOR....

In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying...

Dear Friends of Numbers: This little book is for you. It should o?er an exquisite int- lectual enjoyment, which only relatively few fortunate people can experience. May these essays stimulate your curiosity and lead you to books and articles where these matters are discussed at a more technical level. I warn you, however, that the problems treated, in spite of - ing easy to state, are for the most part very di?cult. Many are still unsolved. You will see how mathematicians have attacked these problems. Brains at work But do not blame me for sleepless nights (I have mine already). Several of...

Paulo Ribenboim behandelt Zahlen in dieser aussergewohnlichen Sammlung von Ubersichtsartikeln wie seine personlichen Freunde. In leichter und allgemein zuganglicher Sprache berichtet er uber Primzahlen, Fibonacci-Zahlen (und das Nordpolarmeer ), die klassischen Arbeiten von Gauss uber binare quadratische Formen, Eulers beruhmtes primzahlerzeugendes Polynom, irrationale und transzendente Zahlen.

Nach dem grossen Erfolg von Die Welt der Primzahlen" ist dies das zweite Buch von Paulo Ribenboim, das in deutscher Sprache erscheint.

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This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples. The introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields. Part one is devoted to residue classes and quadratic residues. In part two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, inertia and ramification of ideals. part three is devoted to Kummers theory of cyclotomic fields, and includes...

Fermat's problem, also ealled Fermat's last theorem, has attraeted the attention of mathematieians far more than three eenturies. Many clever methods have been devised to attaek the problem, and many beautiful theories have been ereated with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topie is presented in the form of leetures, where I survey the main lines of work on the problem. In the first two leetures, there is a very brief deseription of the early history, as well as a seleetion of a few of the more representative reeent results....