Customer Reviews

The New Book of Prime Number Records

5 star:		(4)
4 star:		(0)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

This is a fascinating and arguably essential book for mathematically-sophisticated people who want a rigorous overview of prime numbers and related aspects of number theory. It's widely cited in other books, which is how I found it.

The title sounds like it's just a book about the largest prime numbers, but it's far, far more than that, including proofs of virtually all prime number theorems, a great many algorithms for generating primes, computational aspects of generating and testing primes, cryptographic applications, and much more.

The author is a skilled mathematician who is able to write about advanced mathematics in a remarkably clear, even jovial style.

However: the Amazon description says it is "thoroughly accessible to everyone". Utter nonsense.

This quote is actually a truncated version of one on the back of the book, which says it is "...thoroughly accessible to everyone with some mathematical education".

Well, "thoroughly accessible" and "some mathematical education" are relative terms. If you think it means you will understand this book if you took 3 or 4 years of high school math -- algebra, geometry, trig, pre-calculus -- you will be rudely shocked. Even a year or two of college math classes may be insufficient.

While you don't necessarily need an undergraduate degree in mathematics to appreciate this book, it wouldn't hurt -- it's used in graduate math courses.

But if you have the interest, and sufficient background, you should definitely buy this book. And by all means, consider a used copy -- I paid about $20 and it was fine. I think the large number of (barely) used copies available on Amazon has something to do with the number of people who bought it without realizing what they were getting into. If you have the requisite background, you would not let this book go.

Ribenboim wrote a book called the Little Book of Big Primes which came out in 1991 and was written for general audiences. This book and the revised edition of the Little book are of personal interest to me because my father's work in 1939 gets cited a lot with respect to the Carmichael numbers. This motivates me to read through the book in more detail that some of the other reviewers. But as Cube Boy mentioned this is not just about records. It is fundamentally an advanced book on number theory for mathematicians with many of the same topics as are in the Little Book but at a hogher level with more mathematical detail and with new results that came about between 1991 and 1996. I have also reviewed the second edition of the Little Book and I comment more about the prime records and my father's contributions there. Also the second edition in 2004 has new records as the computer is able to verify that larger and larger numbers are prime.

This book is a must have for anybody interested in number theory. It contains a treasure trove of solved and unsolved problems in number theory and records related to them and is highly entertaining.

I've given this book 5 stars without ever having laid eyes on an actual copy or held one in my hand, for the following reasons:
(1) I looked at pages from this book with Amazon's viewer , and saw an impressive table of contents, an impressive excerpt from the first few pages, and an IMPRESSIVE bibliography of about 76 pages!
(2) The two earlier reviewers who seemed to know what the material entails, and who (I presume) had actually read the book, gave it 5 stars.
(3) I came to Amazon's listing of this book because I had seen SO MANY references to it from other authors who were knowledgeable in this field, both in their books and in articles on the Web.

I don't usually review works I haven't read, but felt I had to add my two cents worth to this discussion after I read the trash contained in the review by the person who obviously has no mathematical talent, said as much, and then proceeded to pan the book based on what are either outright misconceptions or metaphysical ramblings. I've got my doubts that he actually read that much of it either! I felt compelled to right the wrong done by his low rating. The funny thing, his rating would have applied to ANY OTHER book on this subject written by a competent person.

For what it's worth, I'm a college-level math instructor, have a master's degree in math, have interest in this subject but don't specialize in it, would like a copy of this book (as well as a whole lot of other books), but on my limited book budget I'm going to hold off until I can find a much cheaper used copy (as one reviewer implied were available, though I didn't see such a bargain on the current listings).

Yet again a book on primes fails to address a serious issue on the subject. One of the things that bothered me about primes is how messy they are. From the perspective of where they are on the number line one can't help but get the feeling that any equation related to their distribution is going to be ugly. Maybe I am a sucker for simplicity- just call it an eye for elegance!

Taking a look at the math culture's definition of a prime we find something like: "..a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself." Oh how boring! Of course the mathematicians tell us that primes build all the other numbers. Digging around one will find this formal statement called the fundamental theorem of arithmetic. It says, "every natural number greater than 1 can be written as a unique product of prime numbers." It appears to be very, very important to mathematics- afterall, it is the fundamental theorem of arithmetic!

I must admit I didn't investigate the prime number sequence at all other than taking a quick peek at the first 100 primes. Instead, I became intensely focused on the two related definitions given above. Take a look at the words in the definition and convince yourself which words convey the most "action"- the meat of the definitions so to speak. I came up with "natural number divisors" and "unique product." Now, I must say right away that I failed calculus II so I do not profess to be a brilliant mathematician (don't worry, I took the class again with a different professor and got an passing grade). There is one thing that I do know about math and it is this: multiplication is just repeated addition.

So, I wondered what would happen if the math culture rewrote the fundamental theorem of arithmetic without using the word "product." Wouldn't that be cool- a simplified version of the definition! Maybe... just maybe... we might find some new way to think about prime numbers and make some progress on the stubborn topic.

Personally, I believe that a number which is "prime" is just highlighting a side effect of short-cut addition. We have to have short-cuts otherwise we humans would count to each other when we simply wanted to say "I'll pay you 25 copper coins to feed my camels." Think about the axioms of arithmetic. List them on paper and then erase the ones related to multiplication and division. Now, tell me what a prime number is! I feel that we have been duped by the math community at large because they told us for so long that primes are super important- even godly. I challenge everyone to go back to the basics for the sake of progress! (I know you're just as tired of the centuries-old unsolved prime number mysteries)