Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory

The authors of this book report that it is intended for use in a problem-solving course for undergraduates. It gives many techniques for solving equations and inequalities, but also provides the reader with the theory behind the techniques. The book is divided into three sections.

The first section deals with Algebraic Identities and Equations. Many of these are of a combinatorial nature.

Determine the value of the sum 1 x _nC₁ + 2 x _nC₂ + ... + n x _nC_n.

There is a derivation of Bernoulli's formula for the sum of the kth powers of the first n positive integers. There are many problems involving polynomials.

Show that the polynomial (1 - x + x² - ... - x⁹⁹ + x¹⁰⁰ ) (1 + x + x² + ... + x⁹⁹ + x¹⁰⁰) has zero coefficients for all odd powers of x.

Find those nonzero polynomials F(x) for which F(x² ) = (F(x))².

The book makes use of symmetric polynomials to solve some equations (and inequalities in the next section).

Let a and b be two distinct roots of the equation x⁴ + x³ = 1. Show that their product ab is a root of the equation x⁶ + x⁴ + x³ - x² - 1 = 0.

Construct a polynomial whose zeros are the squares of the zeros of the polynomial x³ - 2x² + x - 12.

There are problems involving systems of equations, linear and nonlinear. (Some of the nonlinear systems are solved using symmetric polynomials.) There are problems involving irrational equations (equations with radicals). The section finishes with some applications of complex numbers, such as using de Moivre's Theorem to derive combinatorial results. In fact, one of the problems culminates in a proof that the sum of the reciprocals of the squares is equal to pi²/6.

Chapter 2 deals with Algebraic Inequalities. Many methods are presented for solving inequalities. In fact, there are so many, that the student (and--perish the thought--the instructor) may find themselves overwhelmed. The chapter starts simply. For example,

Show that the eighth root of 8! is less than the ninth root of 9!.

Which is larger: 2⁷⁰⁰ or 5³⁰⁰?

But the difficulty soon grows.

For arbitrary positive real numbers a, b, c, show that (a^a b^b c^c)² >= a^{b + c} b^c+a c^a+b.

Show that if a > 1, then 1/(a - a ^-1) > 2/(a² - a^-2) > 3/(a³ - a^-3) > ...

For a positive integer n, if A_n denotes 3³3^... 3 (where n threes are used) and B_n similarly denotes 4⁴4^... 4 , which is larger, A_n or B_{n -1}?

Some of the standard results are presented, such as Cauchy's Inequality, the Arithmetic-Geometric Mean Inequality, and Chebyshev's Inequality. This chapter is considerably more technical than the other two. If one can make it through the material without getting bogged down by the minutiae, then they will almost certainly find themselves with a few more problem-solving tools.

The third chapter (in this reviewer's biased view) is the most interesting: Number Theory. After presenting the basics of divisibility, the Euclidean algorithm, greatest common divisor, least common multiple, and primes, the authors start to give some interesting problems.

Show that 5²⁰ + 2³⁰ is composite.

Show that if ab = cd for any natural numbers a, b, c, and d, then aⁿ + bⁿ + cⁿ + dⁿ is composite for any natural number n.

The book introduces congruences. The Euler Phi function is discussed, along with Fermat's Little Theorem and Euler's Theorem.

Show that for any prime p, infinitely many numbers of the form 2ⁿ - n (where n is a natural number) are divisible by p.

We then return to congruences, dealing with systems of congruences, the Chinese Remainder Theorem, and higher-order congruences.

Solve the congruence x² - 3x - 10 = 0 (mod 49).

The authors spend a great deal of time talking about Diophantine equations.

Linear: Find all integer solutions to 91x - 28y = 35

Higher order: Find all integer solutions to (x - y)² = x + y.

Exponential: Find all integer solutions to 3^x = 4y + 5.

And many others: Find all integer solutions to 1! + 2! + 3! + ... + x! = y².

The method of reductio ad absurdum is presented, and techniques are given to show that a Diophantine equation has infinitely many solutions (even if they can't be explicitly described). There are some problems on integer- and fractional-parts of a number, base representations, and repunits. Some problems are given involving Dirichlet's Principle (aka the Pigeonhole Principle).

Show that from among any fifteen natural numbers one can choose eight such that their sum is divisible by 8.

Suppose that the product of 48 distinct natural numbers is divisible by exactly ten primes. Show that one can choose four of these 48 numbers such that their product is the square of an integer.

The remainder of the chapter deals with polynomials and their properties, including Eisenstein's Irreducibility Criterion.

Show that a polynomial F(x) with integer coefficients that takes on the values 1 or -1 at three different integers has no integer zero.

There is one error in the text (this may be an error in the original manuscript or one that came about in the translation) which occurs in section 4.11 of chapter 3. In the Consequence at the beginning of this section, the hypothesis which reads

"If a₁, a₂, ... , a_s are all integers such that 0 <= a_i < m and F(a_i) = 0 (mod m)"

should read

"If a₁, a₂, ... , a_s are all of the integers such that 0 <= a_i < m and F(a_i) = 0 (mod m)".

There is no shortage of problems and examples here. This book contains over 300 examples and then over 700 exercises for the reader. The authors have even included a fourth chapter with hints and answers to the exercises. So, even if the solution isn't given in the back, at least the reader will have some hint about which direction to start looking.

The book provides a nice introduction into many of the problems-solvers' "tricks of the trade." There is a lot presented here, so the reader (especially an undergraduate) may want to take the following approach: read just a few pages at a time, work out some of the exercises, then take some time to digest the material before going on.

Donald L. Vestal is an Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at vestal@griffon.mwsc.edu.