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 START TRANSACTION WITH CONSISTENT SNAPSHOT; /* 108 = 23b8535ad765882df043ed3243465ee2