Peter Gustav Lejeune Dirichlet (1805–1859) was a German mathematician so careful and so clear that many people say he invented the modern idea of what a proof should look like. He was gentle, a little shy, and famously reluctant to write letters home — legend has it that when his first son was born, he telegraphed his father-in-law nothing but the news that 2 + 1 = 3.

The stroke of genius

Dirichlet proved one of the loveliest results in number theory: that every arithmetic progression whose first term and step share no common factor contains infinitely many primes. To do it he built powerful new tools — the infinite sums now called Dirichlet series — welding calculus onto counting primes and helping launch the whole field of analytic number theory. His name is also stamped on the "Dirichlet function", the "pigeonhole principle" (which he called the drawer principle), and boundary conditions used in physics.