At the start of this chapter we asked three questions about whole numbers, and promised a fourth theme running underneath them: how to prove things. It is time to see how far nine STEP questions have taken us.
1. How many primes are there, and how are they spread out?
In Section 1.1 we proved that there are infinitely many primes without finding a single new one. We built an infinite family of numbers, the Fermat numbers, that never share a factor, so each one needs a prime of its own. It is a proof about structure, not size.
In Section 1.2 we went further and showed that the product of all the primes up to \(n\) is less than \(4^n\). That was our first quantitative statement: it limits how densely the primes can be packed. Taking logarithms, it says \(\sum_{p \leq n} \ln p < 1.39\,n\), while the Prime Number Theorem says the true value is about \(n\). Our STEP-level argument was already in the right ballpark.
What we have not done is find exactly how the primes are spread out. That is controlled by the Riemann zeta function, and making it precise is the Riemann Hypothesis, the most famous open problem in mathematics. You now know enough to understand what it is about.
2. How can you prove that a number is not a fraction?
We found three quite different ways.
Take the smallest one (Section 1.3). If \(\sqrt2\) were a fraction, there would be a smallest positive integer \(k\) making \(k\sqrt2\) whole; but then \((\sqrt2 - 1)k\) would be a smaller one. The same argument handled cube roots.
Trap an integer between 0 and 1 (Section 1.4). If \(e\) or \(e^2\) were a fraction, multiplying by a factorial would produce a whole number lying strictly between \(0\) and \(1\). This is the idea behind Hermite’s and Lindemann’s proofs that \(e\) and \(\pi\) are transcendental.
Find an impossible divisibility (Section 1.5, exercise 3). Showing there is no rational point on \(x^2 + y^2 = 3\) is the same as showing \(\sqrt{3 - x^2}\) is never rational when \(x\) is.
And we learned how much remains unknown: nobody can yet prove that \(e + \pi\) is irrational.
3. Which equations have whole-number solutions?
Sometimes the answer is none, and Section 1.5 showed Fermat’s way of proving it: any solution would produce a smaller one, for ever. Working modulo a well-chosen prime told us which divisibility to exploit.
Sometimes the answer is yes, and here is a machine for producing them. In Section 1.6 the complex numbers turned sums of two squares into a multiplication rule. In Section 1.9 a tangent line to a cubic curve produced \(20^3 = 17^3 + 14^3 + 7^3\).
And sometimes the right move is to change the number system. The Gaussian integers made sums of two squares easy. But Section 1.7 gave a warning: in a system that looks perfectly reasonable, numbers can factorise into “primes” in two different ways. That warning, ignored in 1847, led to the invention of ideals and modern algebra.
Section 1.8 added a twist of its own: the fractions, which seem far more numerous than the whole numbers, can be listed one by one. Infinity has more structure than it first appears.
4. A toolkit of proof
Here are the methods you have used in this chapter. Every one of them will appear again and again in a university mathematics course.
| Method | Where you used it |
|---|---|
| Induction and strong induction | Fermat numbers (1.1); products of primes (1.2) |
| Well-ordering, minimal counterexample | \(\sqrt2\) and \(\sqrt[3]{2}\) (1.3); listing fractions (1.8) |
| Infinite descent | Impossible equations (1.5); coprime pairs (1.8) |
| Proof by contradiction | Throughout, especially 1.3 and 1.4 |
| Working modulo \(n\) | 1.5; 1.7 |
| Bounding and squeezing | 1.2; 1.4 |
| Building a bijection | 1.8 |
| Constructing a counterexample | Non-unique factorisation (1.7) |
| Changing the setting | Complex numbers (1.6); geometry (1.9) |
If you look back at the questions, you will notice that the STEP examiners often gave you the first step: “by considering the binomial expansion…”, “by considering the complex numbers \(2 + \mathrm{i}\) and \(10 + 11\mathrm{i}\)…”. At university that help gradually disappears, and finding the right first step becomes the skill. The best preparation is exactly what you have been doing: when you finish a problem, ask why the hint worked, so that next time you can think of it yourself.
Where next
Several threads from this chapter continue elsewhere in the book.
The approximation of \(e\) by fractions returns in Section 2.9, and the Riemann zeta function in Section 2.4, where \(\zeta(2) = \pi^2/6\).
Roots of unity, which appeared in Lamé’s failed proof, are the subject of Chapter 5, and Gauss’s construction of the regular 17-gon, promised in Section 1.1, is Section 6.2.
The four-square version of the identity in Section 1.6 appears in Section 7.8, via quaternions.
Harold Davenport, The Higher Arithmetic: the best single introduction to elementary number theory.
Aigner & Ziegler, Proofs from THE BOOK: Chapters 1–4, 8 and 17 connect directly with this chapter.
Joseph Silverman, A Friendly Introduction to Number Theory: covers almost everything here, and more, at a gentle pace.
Marcus du Sautoy, The Music of the Primes and Simon Singh, Fermat’s Last Theorem: the human stories behind the mathematics.
Number theory has a habit of drawing people in. Its questions are simple enough to ask on the first day, and deep enough to occupy a lifetime. If some of these sections have left you wanting to know what happens next, that is exactly the point.