Dirichlet's theorem on arithmetic progressions

The Dirichlet prime number theorem ( by PGL Dirichlet ) is a statement from the mathematical branch of number theory, which states that an arithmetic progression contains infinitely many primes, if this is impossible for trivial reasons.

In the simplest version of the sentence is: It should be a natural number and a to prime natural number. Then contains the arithmetic progression

Infinitely many primes. In other words: There are infinitely many primes congruent to modulo.

Would not relatively prime and a common divisor, so each follower would be divisible by; but two different prime numbers can not both be divisible by. Therefore, the condition of the relatively prime and necessary.

Each odd natural number, the form, or with a non-negative integer. The Dirichlet prime number theorem states, in this special case that there are two forms of each of infinitely many primes.

Based on the decimal system the theorem states that there are infinitely many primes, respectively, which terminate in the decimal system to a 1, a 3, a 7 and a 9. Generally, one can say: There are two different prime numbers that end in a number system in the same sequence of digits, so there are infinitely many primes further that end in this number system in this number sequence.

In a quantitative version, which follows for example from the tschebotarjowschen tightness set, then the Dirichlet prime number theorem:

With the Euler's φ - function. This statement means that it modulo in a sense are the same number of primes in each of the prime residue classes.