Nuggets of Number Theory: A Visual Approach

The starting point for this book was the author’s observation that many number theory texts contain few figures. He wrote an article for Math Horizons in 2008 that showed ways to use figures (the article was reprinted in the book Biscuits of Number Theory), and eventually expanded it to the present book. The author is probably best known for his many contributions to the series “Proofs Without Words”. The present book has lots of words, but is slanted toward visuals.

The book starts out with figurate numbers (squares, triangular numbers, etc.) that have an obvious picture, but quickly moves to other areas that we usually don’t think of visually. This is not a complete introduction to number theory, but it does cover a good sampling, with something about congruences, Diophantine equations, irrational numbers, perfect numbers, and Fibonacci numbers.

The visual approaches fall generally into two types: variously-shaped arrays of points, that we count in two ways to get an identity, and variously-shaped geometric figures, whose area we calculate in two ways to get an identity. The book often uses a “carpets theorem” that I had not seen before: Suppose two carpets or rugs are laid out in a room, possibly with overlap. Then the total area of the carpets equals the total area of the room if and only if the area of the overlap equals the area of the bare part of the floor. This is true regardless of the shapes of the carpets or of the room; it is a form of the inclusion-exclusion principle.

Sometimes the visual approach is used to get started, and the problem is finished with an algebraic solution. A good example is the Pell equation \(x^2 - d y^2 = 1\) for a square-free \(d > 0\). (I think Pell’s equation is one of the most hard-to-explain parts of elementary number theory, so I was especially pleased to see new treatments here.) The book has clever geometric solutions for \(d=2\) and \(d=3\), but says this becomes more difficult for \(d \ge 5\). Armed with these two examples we are able to generalize the form of the solutions and handle the general case algebraically.

The book’s preface states that the book is not a textbook (it is in the Classroom Resource Materials series). But there are exercises, of moderate difficulty, at the end of each chapter, with brief solutions to all problems in the back of the book.

Another book with a similar approach is Benjamin & Quinn’s Proofs that Really Count. Its organizing principle is counting rather than visuals, but many of the proofs include diagrams. There’s also some overlap in the subject matter (Fibonacci numbers and some number theory), although the proofs are generally different.

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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.