24 Chapter 3

in other words, the numbers of the form p + ei, where i < d. These form an arithmetic

progression of length d and period e. Let me show that these numbers are all relatively

prime. Suppose that q is a prime factor of two of the numbers, p + ei and p + ej, where

i < j < d. It follows that q divides the difference of the two numbers, which is equal to

e( j −i). Since i and j are both less than d and e = d!, it follows that all of the prime factors

of e( j − i) are at most d,soq ≤ d. In this case, q divides e and hence ei, and since it also

divides p + ei, it must be that q divides p, which is prime. So q = p, contrary to our choice

of p to be larger than e.

It had been a long-standing open question, since at least 1770, whether one could find ar-

bitrarily long arithmetic progressions of prime numbers (not merely relatively prime). This

much harder question was finally settled in the affirmative in a celebrated 2004 result of

Ben Green and Terrence Tao. The Green-Tao theorem states that for every natural number

d, there is an arithmetic progression of length d consisting entirely of prime numbers.

Mathematical Habits

Choose variable names well. Choose sensible variable names that help remind you of

their meaning. But also try to follow established variable-naming conventions, when

possible, in order to hook into your readers’ expectations about what kind of quantity

your variable represents. For example, many mathematicians use variables n and m to

represent natural numbers or integers, while p and q are probably prime numbers, the

variables x and y frequently represent real numbers, z is often a complex number, and

f and g are likely functions. Conventions can vary between particular mathematical

specializations.

Use technical words with accuracy and precision. Recognize that mathematical

words often carry extremely precise meanings, far more explicit and detailed than the

meanings conveyed in ordinary language by those words. Use technical vocabulary

strictly to carry these more specific meanings.

Define your technical terms. Provide explicit formal definitions for your mathemat-

ical terms, and use the words in accordance with their definitions. Clarify undefined

terms; do not allow vagueness and ambiguity to slip into your analysis simply because

you have not taken the trouble to define your terms.