The Little Book of Bigger Primes [Paperback]

Paulo Ribenboim (Author)

Book Description

Publication Date: January 8, 2004 | ISBN-10: 0387201696 | ISBN-13: 978-0387201696 | Edition: 2nd

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 Pólya Award of the Mathematical Association of America. He is the author of 13 books and more than 150 research articles. From the reviews of the First Edition: Number Theory and mathematics as a whole will benefit from having such an accessible book exposing advanced material. There is no question that this book will succeed in exciting many new people to the beauty and fascination of prime numbers, and will probably bring more young people to research in these areas. (Andrew Granville, Zentralblatt)

Editorial Reviews

Review

From the reviews of the second edition: P. Ribenboim The Little Book of Big Primes "Everyone has at one time or another taken an interest, however fleeting, in prime numbers . . . Ribenboim, however, is clearly a ‘prime nut,’ and this excellent, good-humored book is written for other (actually or potentially) incurable aficionados . . . it is a masterly presentation . . . This genially reader-friendly tour de force, by a scientist whit an encyclopedic and up-to-the-minute knowledge of the subject, is a wholly admirable addition to anyone’s bookshelf."—AMERICAN SCIENTIST "The book presents a wealth of material and references on fundamental theorems, challenging open problems, and the most recent computational records in a language without secrets." (Zentralblatt für Didaktik und Mathematik, November, 2004) "This book is the substantially expanded and updated second edition of the 1991 ‘Little book of big primes’. … The author’s thorough knowledge of and passion for primes comes through clearly. Most of the book is accessible even to interested amateurs or undergraduate students … . At the same time, a specialist can also find it useful to have so many results, open problems and references collected together." (Gábor Megyesi, Acta Scientiarum Mathematicarum, Vol. 72, 2006) "One could say that Paul Ribenboim created a new genre of mathematical writing with the publication of ‘The Book of Prime Number Records.’. … This together with the author’s charming style, made it a success. The book under review is the abridged version of its successor ‘The New Book of Prime Number Records.’ … The reviewer believes that this book really has the ‘fatal attraction’ its author wishes it to exert." (Ch. Baxa, Monatshefte für Mathematik, Vol. 148 (1), 2006) "This is a book of well documented records. … But this is a book of much more than records, providing as good an introduction as any to the theory of primes … . The first edition of the book appeared in 1991 … . It is here expanded and updated … . Throughout the book history, theory, specific cases and other results are woven into a compact but comprehensive presentation. A most useful resource for both teaching and research into the prime numbers." (Alan Sutcliffe, The Mathematical Gazette, Vol. 89 (516), 2005)

From the Back Cover

A deep understanding of prime numbers is one of the great challenges in mathematics. In this book, 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 Pólya Award of the Mathematical Association of America. He is the author of 13 books and more than 150 research articles. From the reviews of the First Edition: Number Theory and mathematics as a whole will benefit from having such an accessible book exposing advanced material. There is no question that this book will succeed in exciting many new people to the beauty and fascination of prime numbers, and will probably bring more young people to research in these areas. - Andrew Granville, Zentralblatt Ribenboim is a rare creature - an academic who writes in an unstuffy way - and his book gives an overview of what is currently known about primes. - Robert Matthews, The Sunday Telegraph …This genially reader-friendly tour de force, by a scientist with an encyclopedic and up-to-the-minute knowledge of the subject, is a wholly admirable addition to anyone’s bookshelf. - John H. Halton, American Scientist

Product Details

• Paperback: 373 pages
• Publisher: Springer; 2nd edition (January 8, 2004)
• Language: English
• ISBN-10: 0387201696
• ISBN-13: 978-0387201696
• Product Dimensions: 9.1 x 6.1 x 0.7 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #1,015,754 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

6 of 7 people found the following review helpful:

5.0 out of 5 stars entertaining book on number theory, November 24, 2009

By 

Michael R. Chernick "statman31147" (Holland PA) - See all my reviews
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This review is from: The Little Book of Bigger Primes (Paperback)

Number theory is a strange branch of mathematics in my mind. The other branches topology, algebra, analysis, geometry and probability are all built on a foundation where mathematical logic builds theorems from axioms, propositions and lemmas. But to me at least number theory is different. the great results seem to be inspirational and come from brilliance rather than organization and logic. Erdos and Ramanujan are both examples of brilliant minds that made fantastic discoveries seemingly out of thin air and at least in Ramanujan case without much training. Fermat's last theorem may be an exception as it was such a difficult problem that it took hundreds of years and the development of new mathematical structures to solve.

This book is all about primes a very mysterious sequence of number with a very simple property that they are integers with 1 and themself as the only proper divisors. Yet their structure is illusive. There is a theory that tells you for large n approximately how many primes there are less than n. Of course 2 is the only even prime but after if you find a prime when will you see the next one. It could be the next odd integer (forming twin primes) or it may not occur for a long time. Given the long fascination that man has had with numbers and primes in particularly there is still a lot that is not known about the primes. We know there are infinitely many primes, but given a particular very large number is there any quick way to tell whether or not it is a prime. The larger the number the more difficult the task is. This is the second edition of a book called "The Little Book of Big Primes" and is aptly renamed "The Little Book of Bigger Primes". The author is looking at records that can always be broken. That is because when you find the largest prime ever confirmed to be a prime you know that there are more that are bigger. There is a clever proof by contradiction to show that there are infinitely many primes.

This book provides a recording of the largest known primes, when they were found and when the were broken. With modern fast computers and clever methods every few years the record gets broken.

What first attracted me to this book was my brother's discover that Ribenboim references results from my father's 1939 paper "On Fermat's Simple Theorem". In that paper my father produced a technique to generate large primes based on a set of numbers called Carmichael numbers. In 1969 my father who had not worked in number theory for at least 30 years tutored me and some of my Stony Brook classmates in number theory which was difficult to learn from our Hungarian professor. But my fathers explanations were clear and made it seem so much simpler. The next year when I was working at Aberdeen Proving Ground he approached me with an idea. He said that with his knowledge and my computing skills we could publish a paper establishing the largest primes known. The idea was a pleasant one to both of us. Tragically he died in April of 1971 before we even had a chance to discuss the problem.

But now after reading chapter 2 of the first edition of the book I learned that in 1989 Dubner produce a 3710 digit Carmichael number based on a modification of my father's method. The three factors of the Carmichael number are all primes and can be large themselves. This was 50 years after my father's paper and just thirty years after the publication my father still was confident that he could find such numbers and no one else probably would because of the need to have fast computers. Of course the speed of computers in the 1970s pales in comparison to today. So it is not surprising that the number theorist familiar with work like my father's could continue to set records. The first edition of the book appeared in 1991 and the second in 2004. But in 2004 Ribenboim tells us that the largest known general prime number determined by a primality test has 5878 digits. But Dubner has found for a special class of primes called palindromic primes.the number can be expressed as 10 to the power 39026 + 4538354x10 raised to the power 19510 + 1. This prime has 39027 digits. Since 2004 there have been a number of larger primes found mostly of the Mersenne variety and i believe they are up to or past 1 million digit primes.

Pure mathematicians are proud of the fact that their work is theoretical and has no know applications. it is mathematics for it pure beauty alone. If there was ever a branch of mathematics that was consider totally pure it was surely number theory. But alas that is no longer true in this modern age of encryption very large primes can be used to create encryption codes that are extremely difficult to break.

Ribenboim writes in a very pleasing style and makes complex things seemingly simple. He also has that knack with his other books particularly those that explain Fermat's last theorem.

I was extremely gratified to find someone other than his children who finds my father's work in number theory to be interesting and useful!

1 of 1 people found the following review helpful:

5.0 out of 5 stars Better than the rest, November 20, 2010

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hot4hypatia (29.48 N , 98.51 W) - See all my reviews

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This review is from: The Little Book of Bigger Primes (Paperback)

I bought this used on Amazon and I am very impressed with what the author has achieved. He gives serious results together with lots of history and anecdotal facts about prime numbers. I also own the more recent 'Prime Curios' book and I think that the little details about primes the author included in this book are qualitatively better.

3 of 12 people found the following review helpful:

4.0 out of 5 stars This is an obsolete edition., June 9, 2000

By 

John Lord - See all my reviews

This review is from: The Little Book of Big Primes (Paperback)

Hasn't this book been replaced by the 2nd edition, 1998, same publisher?