# Dirichlet’s theorem (analytic number theory)

The Dirichlet ‘s theorem in the analytic theory of numbers , on the infinite number of primes rational in arithmetic progressions is a proposition, proven by the mathematician Pedro Gustavo Lejeune Dirichlet . This theorem about the distribution of rational prime numbers , was intuited by the Prussian mathematician Gauss and proved in 1837 by Dirichlet, and honors the memory of the latter.

## Summary

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• 1 The proposition
• 2 Brief proof
• 4 References
• 5 Sources

## The proposition

the arithmetic progression a, a + b, a + 2b, a + 3b, … implies an infinite number of integer silos prime numbers a and b> or they are realative primes [1]

For example, the theorem assures that there is an infinite amount of prime numbers that end in 7, since the numbers that end in 7 form an arithmetic progression (7, 17, 27, 37, …) that is, it is a sequence of numbers of the form a + nd with a = 7 and d = 10, these primes being each other , then their greatest common divisor is 1.

## Succinct proof

The proof of the theorem uses the properties of certain multiplicative functions (known as Dirichlet’s L-functions ) and various results on complex number arithmetic and is complex enough that some classical number theory texts decide to exclude it from their repertoire of proofs.