Summing It Up

From One Plus One to Modern Number Theory

Avner Ash, Robert Gross

• Publisher: Princeton University Press
• Year: 2016
• ISBN:
  9781400880539
• Topic: Matematica
• Subtopic: Teoria dei numeri

The power and properties of numbers, from basic addition and sums of squares to cutting-edge theory We use addition on a daily basis—yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research.Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series—long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+...=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms—the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat's Last Theorem.Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.

PART ONE

Finite Sums

Chapter 1

PROEM

In the interest of allowing the reader to enjoy our book without constantly referring to many other references, we collect in this chapter many standard facts that we will often use in the remainder of the book. A reader familiar with elementary number theory can skip this chapter and refer back to it when necessary. We covered most of these topics in Ash and Gross (2006).

1. Greatest Common Divisors

If a is a positive integer and b is any integer, then long division tells us that we can always divide a into b and get an integer quotient q and integer remainder r. This means that b = qa + r, and the remainder r always satisfies the inequality 0 ≤ r < a. For example, if we take a = 3 and b = 14, then 14 = 4 · 3 + 2; the quotient q = 4 and the remainder r = 2. You may not be used to thinking about it, but you can do this with b < 0 also. Take b = –14 and a = 3, and –14 = (–5) · 3 + 1; the quotient is q = – 5, and the remainder is r = 1. Notice that if we divide by 2, the remainder will always be 0 or 1; if we divide by 3, the remainder will always be 0, 1, or 2; and so on.

If the result of the long division has r = 0, then we say that “a divides b.” We write this sentence symbolically as a | b. Of course, one requirement for long division is that a cannot be 0, so whenever we write a | b, we implicitly assert that a ≠ 0. If the remainder r is not zero, we say that “a does not divide b.” We write that assertion symbolically as a∤b. For example, 3 | 6, 3∤14, and 3∤(–14). Notice that if n is any integer (even 0), then 1 | n. Also, if a is any positive integer, then a | 0. At the risk of giving too many examples, we also point out that 2 | n means that n is even, and 2∤n means that n is odd.

Suppose now that m and n are integers that are not both 0. We can then define the greatest common divisor:

DEFINITION: The greatest common divisor of m and n, symbolically written (m, n), is the largest integer d such that d | m and d | n. If the greatest common divisor of m and n is 1, we say that m and n are relatively prime.

Because all divisors of m are at most as big as m (if m > 0) or –m (if m < 0), we can theoretically list all divisors of m and all divisors of n, and then pick the largest number that is on both lists. We know that the number 1 is on both lists, and there may or may not be any larger number simultaneously on both lists. For example, (3, 6) = 3, (4, 7) = 1, (6, 16) = 2, and (31, 31) = 31. This process would be tedious, though, if we wanted to compute (1234567, 87654321). There is a process called the Euclidean algorithm, which allows one to compute greatest common divisors without listing all of the divisors of both m and n. We will not describe that process here, but we will state and prove one consequence, often called Bézout’s identity.

THEOREM 1.1: Suppose that m and n are not both 0, and suppose that d is the greatest common divisor of m and n. Then there are integers λ and μ such that d = λm + μn.

You can skip the proof if you like. It’s actually a frustratingly incomplete proof, because we aren’t going to tell you how to find λ and μ. Part of what the Euclidean algorithm does is to let you find λ and μ quickly.

PROOF: Let S be the following very complicated set, in which the symbol Z stands for the set of all integers:

In words, S is the set of all multiples of m (positive, negative, and 0) added to all multiples of n (ditto). Because S contains 0 · m + 0 · n, we know that S contains 0. Because S contains m, –m, n, and –n, we know that S contains some positive integers and some negative ...