# Dirichlet, Peter Gustav Lejeune (1805–1859)

Peter Dirichlet was a German mathematician who made significant contributions to number
theory, analysis, and mechanics, and
who is credited with the modern formal definition of a function.
He taught at the universities of Breslau (1827) and Berlin (1828–1855) and
in 1855 succeeded Carl Gauss at the University
of Göttingen but died of a heart attack only three years later. Dirichlet
continued Gauss's great work on number theory, publishing on Diophantine
equations of the form *x*^{ 5} + *y *^{5} = *kz *^{5}. His book *Lectures on Number Theory* (1863)
is similar in stature to Gauss's earlier *Disquisitiones* and founded
modern algebraic number theory.
In 1829 he gave the conditions sufficient for a Fourier
series to converge (though the conditions necessary for it to converge
are still undiscovered).

## Dirichlet's theorem

Dirichlet's theorem states that for any two positive coprime integers, a
and b, there are infinitely many prime
numbers of the form *a*^{ n} + *b*, where
n > 0. This theorem was first conjectured by Karl Gauss and proved by Peter Dirichlet in 1835.