# Paulo Ribenboim

### Born: 13 March 1928 in Recife, Pernambuco, Brazil

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**Paulo Ribenboim**attended the João Barbalho school in Recife, the capital of Pernambuco state in northeastern Brazil, in 1934 and 1935. Recife is an Atlantic seaport containing many waterways, with bridges connecting different parts of the city. In 1936 Paulo's family moved from Recife to Rio de Janeiro, and there he continued his education attending the Anglo-American College which was a private school. He achieved a school record by being awarded five gold medals in each of the years from 1938 to 1942. He attended the Andrew's College from 1943 to 1945.

In 1946 Ribenboim entered the Faculty of Philosophy at the University of Brazil. He wrote his first paper

*Characterization of the sup-complement in a distributive lattice with last element*in 1948 and it was published in

*Summa Brasil. Math*. in the following year. Ribenboim was awarded his Bachelor's degree in 1948 then, following this, he spent a semester as a teaching assistant in the Saint Ignatius Faculty of Engineering which became part of the new Pontificia Universidade Católica in Rio de Janeiro, which is now named the Federal University of Rio de Janeiro. In 1949 he was appointed as a teaching assistant in geometry in Rio de Janeiro, then later that year he was appointed assistant professor at the Centro Brasileiro de Pesquisas Fisicas, again in Rio de Janeiro.

In 1950 Ribenboim was awarded a scholarship to study in Nancy, France, with Dieudonné. He travelled by boat from Rio de Janeiro arriving in Cannes on 22 April before reaching Nancy on the following day. This was an exciting time for Ribenboim for as well as working with Dieudonné he met Laurent Schwartz and Grothendieck while studying in Nancy. During 1951 up to the spring of 1952 he attended lecture courses on Lie groups by Delsarte, algebraic numbers by Dieudonné and the theory of distributions by Laurent Schwartz. In the middle of this mathematical activity Ribenboim married Huguette Demangelle in Nancy on 19 December 1951. His first son Serge Charles was born in 1953.

Ribenboim submitted his second research paper

*Modules sur un anneau de Dedekind*Ⓣ to

*Summa Brasil. Math*. and it was published in 1952. In July of that year he returned from France to Rio de Janeiro where he was appointed to teach calculus at Escola Tecnica do Exercito and also appointed to teach analytic functions in the Centro Brasileiro de Pesquisas Fisicas. In 1953 he was appointed as a lecturer at the University of Brazil in Rio de Janeiro then in August of that year he travelled again to Europe, this time to work with Krull in Bonn, Germany. He studied ideal theory and valuation theory with Krull between August 1953 and August 1956. During this time he attended the International Congress of Mathematicians in Amsterdam in 1954. Ribenboim's big research breakthrough came when he found a counterexample to a conjecture of Krull. He published his result in

*Sur une conjecture de Krull en théorie des valuations*Ⓣ in

*Nagoya Math. J*. in 1955. This became his third publication. After this Ribenboim began to publish rapidly. In 1956 alone he published:

*Sur une note de Nagata relative à un problème de Krull*Ⓣ;

*Un théorème sur les anneaux primaires et complètement intégralement clos*Ⓣ;

*Sur la théorie des idéaux dans certains anneaux de type infini*Ⓣ; and

*Anneaux normaux réels à caractère fini*Ⓣ.

After three years in Germany, Ribenboim returned to Brazil in August 1956 where he was appointed to the new Institute of Pure and Applied Mathematics in Rio de Janeiro. In the same year he was elected to the Brazilian Academy of Sciences, being the youngest member at the time of his election. Perhaps the most remarkable fact about this early election was that he still had not obtained his doctorate. This was awarded by the University of São Paulo in August 1957. In 1958 he was promoted to Research Chief at the Institute of Pure and Applied Mathematics. In the same year he attended the International Congress of Mathematicians in Edinburgh, being the second such congress he had attended. He was the made head of the Mathematics Division of the National Research Council in Brazil and in September 1959 he made his first visit to the United States when he was Visiting Associate Professor at the University of Illinois. This visit was funded by a Fulbright Fellowship. While in Urbana in 1961 his second son Eric Leonard was born.

In 1962 he had a summer appointment at the University of Michigan in Ann Arbor but, after this, his United States visa could not be renewed. He was then appointed as associate professor of mathematics at Queen's University in Kingston Ontario, Canada. His tenure at Queen's University was confirmed in 1964 and in the following year he was promoted to full professor.

Ribenboim has published a remarkable collection of research papers, monographs, graduate and undergraduate texts, and popular widely accessible mathematics books. A bibliography in [1] contains 262 items beginning with his work on valuations and valuation rings, continuing with a remarkable collection of books, up to his latest papers on ultrametric spaces. Let us look at some of the books which he has published. His first book was

*Théorie des groupes ordonnés*Ⓣ published in 1963. This is a:-

Several texts followed in fairly quick succession:... clearly written introduction to the theory of abelian ordered groups, assuming only an elementary knowledge of abelian group theory and topology.

*The Riemann-Roch theorem for algebraic curves*(1965) and

*Linear representation of finite groups*(1966). Ribenboim, in the Preface to the second of these writes:-

In 1965 Ribenboim published his bookThese notes cover the second part of a course to fourth year undergraduate honours students in mathematics at Queen's University, given in1964... Apart from the arrangement and selection of material, there is little that is original.

*Théorie des valuations*Ⓣ consisting of lectures he gave in 1964 at the Séminaire de Mathétiques Supérieurs in Montréal.

In 1966 Ribenboim spent two months at Harvard working with Zariski. In 1966-7 he spent a sabbatical at the North-Eastern University in Boston. During this time he attended lectures by David Mumford on algebraic geometry. He also wnt to lectures by J Ax and these formed the basis of his next book

*La conjecture d'Artin sur les équations diophantiennes*Ⓣ published in 1968.

*Rings and modules*published in 1969 was designed for instructors rather than for students. Ribenboim points out in the Preface that he considers preparation of exercises as well as expansion of material to be roles of a competent instructor. In 1972

*L'arithmétique des corps*Ⓣ was published. Jacobowitz, in a review of the work, writes:-

Also in 1972 he publishedThis is a text suitable for an unorthodox course in algebraic number theory. Instead of the customary material on ideal factorization and unit theorems, the reader will find such topics as p-adic logarithms, the Witt ring, infinite Galois theory, ordering of number fields and diophantine dimension(including Terjanian's celebrated counterexample). The author possesses an excellent feeling for the precise amount of detail needed in a proof, which makes much of the book accessible even at the advanced undergraduate level. There are many exercises of a theoretical, rather than routine computational, nature which complement the text. In short, the author has written an attractive exposition of his subject.

*Algebraic numbers*. The book is divided into 13 chapters, the first four of which provide background material. The next six chapters cover algebraic integers, integral basis, discriminant, the decomposition of ideals in Dedekind domains, the norm of an ideal, finiteness of the class number, the structure of the group of units, the ramification index and the degree of inertia, discriminant and different. Chapters 11 and 12 treat more specialised topics. The first of these is devoted to ramification theory in Galois extensions and the second to a proof of the theorem by Kronecker and Heinrich Weber on the abelian extensions of the field of rational numbers.

Many people will think of Ribenboim as a writer of superb number theory books. He began to write about this topic, one which clearly had an enormous fascination for him, in 1979 with his famous text

*Thirteen lectures on Fermat's last theorem*.

Further wonderful books of number theory have excited many students and turned them on to mathematics.

*The book of prime number records*published in 1988 has six chapters entitled:

How many primes are there?;

How to recognize whether a natural number is prime;

Are there functions defining prime numbers?;

How are the primes distributed?;

What special kinds of primes have been considered?;

Heuristic and probabilistic results about prime numbers.

A reviewer's comment "This book is well written with much wit" certainly applies to this book, but could equally apply to all of Ribenboim's books. Another classic is How to recognize whether a natural number is prime;

Are there functions defining prime numbers?;

How are the primes distributed?;

What special kinds of primes have been considered?;

Heuristic and probabilistic results about prime numbers.

*The little book of big primes*(1991) which is an abridgement of the book just described. It has a sequel:

*The little book of bigger primes*(2

^{nd}edition, 2004). Further number theory books are

*Catalan's Conjecture*published in 1994,

*The new book of prime number records*(1995) and

*Fermat's last theorem for amateurs*(1999).

More recent books by Ribenboim are:

*The theory of classical valuations*(1999);

*My numbers, my friends*(2000);

*Classical theory of algebraic numbers*(2001) In the Preface to the first of these Ribenboim explains the what he intends to study:-

In addition to writing many books, Ribenboim has edited a number of 'Collected Works' of his fellow mathematicians. For example, among the mathematicians included in this archive, he has edited the Collected Papers of Satoshi Suzuki, Wilhelm Ljunggren, Norman Alling,Pierre Samuel, Giacomo Albanese, Leo Moser, Wolfgang Krull and Mario Fiorentini. He has also edited the works of Peter Roquette, Trygve Nagell, Kustaa Inkeri, Ruggiero Torelli and Wayne McDaniel. His own seven-volume collected papers appeared in 1997 and a new edition will be issued soon.Kürschák formally introduced the concept of a valuation of a field, as being real valued functions on the set of nonzero elements of the field, satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt, and others developed this theory. These are the classical valuations which are the object of this book. Krull extended the concept of a valuation of a field, by allowing the values to be in any totally ordered abelian additive group. Such valuations are often called Krull valuations; the classical valuations are special Krull valuations. In this book we shall not study Krull valuations.

Ribenboim has received a wide range of honours for his outstanding contributions to research and for the outstanding books he has written. He was elected a fellow of the Royal Society of Canada in 1969. Other honours have included: an honorary doctorate from the University of Caen, France, in 1979; receiving the Rosette of the Royal Society of Canada in 1982; and a prize for excellence in research from Queen's University in 1983. He received the Mathematical Associaton of America's George Polyá Award in 1995.

Let us end by quoting from two reviews of

*My numbers, my friends*, a book which both in title and content say much about Ribenboim's mathematical loves. An extract from a review of the text by F Beukers gives a good impression of the contents:-

The final extract is from a review by Marvin Schaefer:-This is a book by a man who loves numbers. In10long essays the author deals with number theoretic properties of numbers which have captivated both professional and amateur mathematicians. Examples of such numbers are Fibonacci numbers, class numbers, prime numbers, consecutive powers, powerful numbers and1093. Besides that we find chapters on prime-producing polynomials and transcendence results. The text is written on the level of undergraduate students. The choice of topics is of course personal. But in several of them, such as powers in recurrent sequences, the author has made a number of contributions. And on other topics included, the author has written complete books, such as on Catalan's problem, prime records and Fermat's Last Theorem.

What can I say? I am a fan of Paulo Ribenboim, and I avidly look forward to reading his new books. His other works include 'The Little Book of Big Primes'(1991), '13Lectures on Fermat's Last Theorem'(1995), 'The[New]Book of Prime Number Records'(1998), and 'Fermat's Last Theorem for Amateurs'(1999). His writing is almost always clear and warmly wry. Most importantly, he takes special pains to make certain that he includes the most recent results and an extensive bibliography. So when 'My Numbers, My Friends' came out, I was excited. And this book more than rewarded my anticipation! Ribenboim is now retired, and treats us to a delightful survey of a few of his favourite numbers ... This book has a number of wonderful and wryly-written passages. Its coverage is vast: although much of the material in the lectures can be found in number theory textbooks, Ribenboim has integrated and consolidated so much related material from the literature that each lecture sparkles from its new treatment.

**Article by:** *J J O'Connor* and *E F Robertson*

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