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Figurate Numbers

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MAA Review

Figurate Numbers is a delightful, well-written text, written by research mathematicians but flexible enough to be used as a class textbook in lower undergraduate, upper undergraduate or beginning graduate courses.

Lower-level undergraduate course: This book is useful as a supplemental textbook in lower undergraduate courses for non-math majors. Not all parts of it are accessible to such students, but the first few chapters have ample historical anecdotes, non-algebraic proofs, and plenty of exercises.

Upper-level undergraduate course: This book could be used as a textbook for a number theory course. The book does not have all the usual number theory material. For example, sections on congruences, standard number theoretic functions (e.g. Euler’s phi), and quadratic reciprocity are absent. But the text covers many number theoretic topics (Diophantine equations, recursions, generating functions, geometrical approaches), mentions many classes of special numbers (Catalan, Stirling, Bell, Bernoulli, binomial, Cunningham and Sierpinski numbers), and has abundant exercises with the right level of challenge for upper-level undergraduates. In other words, this book provides an alternative to the standard undergraduate number theory course.

Graduate course: This book could be used for a beginning graduate course provided the course is supplemented with the reading of papers (to which beginning graduates should be exposed anyway). Although this book does not cover certain topics, it does mention them and use them in proofs, making the text a sort of springboard to bring in these other topics. For example, Legendre’s symbol, Jacobi’s symbol, Dirichlet’s theorem regarding primes in an arithmetic progression, binary quadratic forms, Minkowski’s theorem arise naturally in proofs in the text. The book even touches on transcendence theory when discussing the Catalan conjecture that 2³ and 3² are the only non-trivial powers that differ by 1.

The book studies the figurate numbers, such as the triangular, square, and polygonal numbers. Beyond the standard plane figurate numbers, the authors treat centered polygonal numbers, 3-dimensional figurate numbers, multi-dimensional figurate numbers, Fermat’s polygonal number theorem. These are followed by a general chapter on related number theory topics, including Pascal’s theorem, Pythagorean triples, Diophantine equations, perfect numbers, Mersenne, Fermat, Fibonacci and Lucas numbers, palindromic numbers, Waring’s problem and magic constructions.

There are at least five features that make this book suitable for lower undergraduate or upper undergraduate courses:

• Proofs: The authors rarely given a single proof of a theorem. Rather, theorems might be given 3 or 4 proofs. Proof methods include induction, geometry, generating functions, and summation formulas.
• Exercises: The book has 155 exercises with 155 worked out solutions. Furthermore, if one counts the exercises with multiple parts, there are an additional 150 exercises. Some exercises involve routine proofs, but others are exploratory, offering good challenges for upper undergraduates and beginning graduate students.
• Historical material: Figurate numbers date back to antiquity. The authors include this historical background in their text. This material will be of special interest to non-mathematics majors in an introductory liberal arts course.
• Structure: Despite the wealth of material, topics, and exercises, the book has the lean and lively look. It has only 400 pages (excluding exercises) and 43 sections.
• Tables and figures: Throughout the book there are ample figures and tables of special numbers and functions.

This is a delightful book on a delightful topic. Even if you don’t want to use this book for a class, you should still buy it for yourself.

Russell Jay Hendel (RHendel@Towson.Edu) holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.

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