From Great Discoveries in Number Theory to Applications

This is somewhere between a popular math book and a text in elementary number theory. It reminds me in many ways of Ore’s Number Theory and Its History. The organization is more that of a popular math book, with topics grouped by a sort of free association of ideas rather than by a logical structure. It also has more applications than is usual in either kind of book.  Apart from that it is very conventional and has the theorems and proofs that you would expect. The book is slanted toward the number theory of particular kinds of integers, such as primes, Fibonacci numbers, polygonal numbers, pseudoprimes, and Mersenne and Fermat primes.

 

The term “Great Discoveries” in the title is probably an exaggeration unless we are thinking about things that were great discoveries 2,000 years ago. The book does cover a number of newer discoveries, but the applications all depend on older mathematics, using properties of the primes and of relatively-prime pairs of numbers. Still, there is an impressive variety of applications.

 

The centerpiece of the book is Chapter 10, which studies the design of the Prague Astronomical Clock. The clock was installed in its original form in 1410 and has been added to and revamped many times since then. Much of the chapter focuses on the “bellwork” which is the mechanism for ringing the hours. Getting the correct number of rings for each hour depends on a clever arrangement of two gears; the arrangement is abstracted and studied here as “Šindel sequences”, named after Joannes Andreae (called Šindel), a Czech astronomer who either designed or inspired the original clock. The book also takes a brief look at the main clock mechanism, that includes large gears of 365, 366, and 379 teeth.

 

The book ends with two more chapters of applications. Chapter 11 deals specifically with the uses of prime numbers and most of the applications deal with codes of various sorts, including check digits, encryption, and hashing. There’s also some material on random number generators and how a drawing was digitized for transmission to possible extraterrestrial civilizations in 1974.  Chapter 12 is more varied and deals with error-correcting codes, the five Platonic solids, Latin squares, the analogy of the Eight Queens Problem for prime-sided chessboards, and several geometric problems.

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Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal website is allenstenger.com. His mathematical interests are number theory and classical analysis.