The Little Book of Bigger Primes

This book is the little sibling of Ribenboim’s The New Book of Prime Number Records. It is smaller and newer than its sibling but has the same general organization. Both books are slanted towards areas that admit “records”, such as Mersenne primes, where only finitely many are known and there is an ongoing effort to find bigger ones. With computers in the mix, records are broken frequently and any book of records quickly becomes obsolete. Chris Caldwell at the University of Tennessee at Martin maintains The Prime Pages, with lists of the biggest primes of many types and background information.

 

Happily the present book includes not only records, but the theorems and algorithms that are used to seek new records or support the existence of new records to be sought, and these have a much longer shelf life. For example, the present book lists the 38 known Mersenne primes (51 are are known in early 2020), but also states and proves the Lucas–Lehmer test that is used to find new Mersenne primes, and describes the GIMPS project (Great Internet Mersenne Prime Search), a collaborative effort to find these primes. The book includes many proofs, most of them short and not requiring much background; for the more difficult theorems, it states them and gives references. The bibliography is impressive and huge, at 45 pages in a 350-page book.

 

The book covers a large number of different prime-related topics. Most of these are covered in more detail in specialized texts, but the overviews here are still very handy. There are a few topics that are covered in ore detail here than in other books; these include the many kinds of pseudoprimes, primes of special forms (such as Mersenne or Fermat primes), prime values of polynomials, and the size of the gap between consecutive primes. This book is where Firoozbakht’s Conjecture, that the sequence \(p_n^{1/n}\) is strictly decreasing for \(n \ge 2\), where \(p_n\) is the \(n\)th prime, was first published. The book was published in 2004, so it omits some important new results, such as the Green–Tao theorem (2004) that there are arbitrarily long sequences of primes in arithmetic progression, and the Zhang theorem (2013) that there are infinitely many pairs of primes that differ by less than a certain constant (a major advance in the twin-prime problem). 

 

The two Ribenboim books are very similar, and their Tables of Contents are almost identical. The Little Book is eight years newer and so has more current information. It is condensed compared to its older sibling by omitting some detail of some topics; a few whole topics are omitted, but most omitted topics are obscure. The least-obscure omission is a section on the Waring problem that appears in the Records book. This topic, which is covered in detail there, is not about primes but is of interest to students of primes mostly because the methods of attack are very similar to those for Goldbach’s conjecture. Both books are aimed at researchers. If I could buy only one I would get the Little Book, because it is newer and almost as complete as the other.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.