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Recreations in the Theory of Numbers

Albert Beiler
Dover Publications
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
This is a popular math book that covers most of elementary number theory. It illustrates rather than proves its theorems, with a vast number of numerical examples. I think when it came out in 1964 it was aimed at the intelligent layman and the bright high school student, but those markets have largely dried up today.  There’s a great deal about primes and about primes of special forms such as Mersenne and Fermat, and the related subjects of perfect and amicable numbers. The book covers some easy subjects such casting out nines and representations to different bases and some not-so-easy subjects such as the Euler phi-function, Farey sequences, primitive roots, and Pythagorean triples. Then it gets into some even more difficult subjects such as the Pell equation, binary quadratic forms, and quadratic reciprocity. The book does not prove any theorems, but it does state a large number of theorems that can be expressed by formulas, such as the parametric form of Pythagorean triples. Each chapter has an extensive bibliography, listing both popular and scholarly works.
Many chapters have exercises scattered through them. These usually ask for numbers with particular properties. The last chapter consists of 100 miscellaneous problems. Most of the problems are not very easy. All have answers at the end of the book, although usually just the final answer and not an explanation.
One especially interesting chapter even today is Chapter XXI on mechanical methods of factoring, which includes a long narrative on the Lehmers’ custom-built factoring machines.
Two more modern books, which have a similar flavor but are aimed at mathematicians, are Ribenboim’s The Little Book of Bigger Primes and The New Book of Prime Number Records.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web site is His mathematical interests are number theory and classical analysis.


Presentation at Court
  Divisors for Diversion
  Just Between Friends
  Invention of the Master
  Open Sesame
  The Cup of Tantalus
  Digits--and the Magic of 9
  Scales and Discords
  Cycling Towards Infinity
  11111 ... 111
  Queer Logs--Back to the Primitive
  The Eternal Triangle
  On the Square
  Farey Tails
  Round the Perimeter
  Ball Games
  Theorema Aureum
  Among the Himalayas
  The Pellian
  The Stone Wall
  Tilts and Tourneys
  The Queen Explains: Solutions to Problems