Primes are the indivisible building blocks of every number

Du Sautoy opens with the foundational fact that every whole number greater than one can be built by multiplying together a unique set of prime numbers, much as chemical compounds are built from elements. This property, known since antiquity and formalized by Euclid, makes primes the atomic units of arithmetic: understanding their distribution is understanding something fundamental about the structure of numbers themselves.

Euclid also proved, with an elegant argument still taught today, that there are infinitely many primes — no matter how far you count, more will always appear. But knowing they never run out says nothing about how frequently they appear or how to predict where the next one lies, and that gap between existence and pattern is what drives the rest of the book.

Du Sautoy frames this tension — infinite in number, yet stubbornly resistant to an exact formula — as the reason mathematicians have spent over two thousand years fascinated by what looks, at first glance, like a simple counting problem.

Takeaway: the deepest mysteries in mathematics often hide inside the simplest-sounding questions.