Review of "Mainly Natural Numbers" by Henry Ibstedt

Review of

Mainly Natural Numbers – a few elementary studies on Smarandache sequences and other number problems, by Henry Ibstedt, Bookman Publishing, Martinsville, Indiana, 2003. 95 pp., $19.95 (paper). ISBN 1-9323901941.

Five stars

 In this book, Ibstedt concentrates on exploring number theory challenges related to several problems, functions and concepts introduced by Florentin Smarandache. Many are based on the Smarandache function

S(m) = n, where n is the smallest positive integer that divides m!

and the Pseudo-Smarandache function

Z(m) = n, where n is the smallest positive integer that 1 + 2 + 3 + . . . + n is divisible by m.

These two functions are easily understandable and provide many new opportunities to investigate the properties of the positive integers. Ibstedt demonstrates and investigates some of those new opportunities.

 His tactics are to present a series of 10 largely distinct short papers, each examining a specific topic. Tables and images are often used to make the specific points.

 Number theory is simultaneously one of the easiest and hardest areas of mathematics. In the tradition of the best work on number theory, Ibstedt solves some problems while developing the background for additional questions for the reader to consider. The level of discourse is well within the range of knowledge of people familiar with basic number theory.