Methods of Solving Number Theory Problems

Number theory, like geometry, is a subject I do not, in my opinion, see often enough given dedicated semesters in undergraduate curricula. Perhaps the role of number theory in modern cryptography has led to an increase in the number of such courses, if not its relevance. When MAA's Committee on the Undergraduate Program in Mathematics (CUPM) formed recommendations to guide mathematics departments in designing undergraduate curricula, it queried Project NeXT mailing Lists members and found something contradicting that hope:

The most persistent dream among our respondents was for math departments and undergraduate programs to be large enough that Number Theory courses could be offered more often and with a greater variety.

From the same study:

Number theory is also famous for having a large number of problems whose difficulty is, shall we say, not obvious to discern on first reading. (This is an issue for anyone teaching the course, or using a particular textbook, for the first time.)

Creativity in Number Theory courses is closely tied to problem-solving and making conjectures. Students should be encouraged to look for patterns and make conjectures and try to verify those conjectures…

Someone trying to offer such a number theory course will find value in this guided exploration of number theory. Not only in a number theory course, but this book can also work for an independent study, preparation for a competition, or part of any course featuring an introduction to proofs. As such, math majors, future teachers, and anyone interested in exploring these ideas on their own will find this work a valuable aid. I myself found reading these orderly, solved problems with detailed solutions (often in about three comparison methods) not only enjoyable but nearly compulsive as noted in margins and applied stick-on tabs. This material telegraphs to a math enthusiast since it is the distillation of a quarter century of collecting as remarked in the preface:

For over 25 years, whenever I spotted an especially interesting or tricky problem, I added it to my notebook along with an original solution. I have accumulated thousands of these problems. I use them every day in my teaching and include many of them in this book.

This notebook basis comes across in some deficiencies, however. Some single-line proofs would be more edifying with a bit more detail. There are also too often grammatical and typographical errors that should have been corrected during editing. While some of these errata merely makes for awkward reading, some are misleading. For instance, the problem to find how many zeroes are at the end of 2010! reads

How many zeros are there in the digital form of the number 2010!?

I did a detailed reading of eighty pages and found six such errors that should have made it to the finished copy, a ratio I feel is consistent throughout this edition. Similarly, some of the material is ordered in a way that should be changed in instruction. Generally, this is acknowledged overtly, such as “The method of solving linear equations like this will be explained in more detail in Chapter 3.”

Topics covered include divisibility, congruences, the Euler phi function, Legendre and Jacobi Symbols, Diophantine equations, and more. Diophantine equations are covered in significant detail, as is Pell’s Equation and related topics. The section “Word Problems Involving Integers” is a gallery of fine problems with thorough solutions. Also comprehensive is a concluding chapter of 110 number theory problems and three dozen pages of detailed solutions. Some more advanced topics not included are p-adic numbers, Bernoulli numbers, and Bernoulli polynomials.

Tom Schulte lives in Louisiana and is a mathematics instructor for Upper Iowa University.