Watkins M. The Mystery of the Prime Numbers 2015

Textbook in PDF format

The Mystery of the Prime Numbers uses an innovative visual approach to communicate some surprisingly advanced mathematical ideas without any need for formulas or equations. The issue of prime numbers acts as a gateway into some truly strange philosophical territory whose relevance extends well beyond mathematics.
An introduction
Numbers and counting and how they’ve taken over the world
How to build the number system: five simple rules that take you out to infinity
Prime numbers: something anyone could have noticed
Prime factors and the most important thing we know about numbers
A philosophical interlude and a journey into space
Addition versus multiplication: they’re surprisingly different things
An infinity of primes: how we can be sure there isn’t a biggest one
Patterns and formulas: the questions everyone seems to ask
Spirals: the central image in this story
The distribution: the curiously random-looking arrangement of primes
Staircases: a useful way to picture the distribution
The deviation: isolating the deviant behaviour of the primes
Harmonic decomposition: breaking everything down into waves
Spiral waves: which no one’s bothered to name properly
Mysterious frequencies: the inevitable cliffhanger ending
Notes
Appendices