Dirichlet's Theorem

Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Dirichlet's Theorem may refer to:

Dirichlet's Approximation Theorem

Diophantine equations

Let $\alpha, x \in \R$.

Then there exist integers $a, q$ such that:

$\gcd \set {a, q} = 1$, $1 \le q \le x$

and:

$\size {\alpha - \dfrac a q} \le \dfrac 1 {q x}$

Dirichlet's Theorem on Arithmetic Sequences

Number Theory, specifically prime numbers

Let $a, q$ be coprime integers.

Let $\PP_{a, q}$ be the set of primes $p$ such that $p \equiv a \pmod q$.

Then $\PP_{a, q}$ has Dirichlet density:

$\map \phi q^{-1}$

where $\phi$ is Euler's phi function.

In particular, $\PP_{a, q}$ is infinite.

Dirichlet's Unit Theorem

Algebraic number theory and ring theory

Dirichlet's Unit Theorem

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.