Dirichlet's Theorem on Arithmetic Progressions

In summary, Dirichlet's Theorem on Arithmetic Progressions, also known as Dirichlet's Theorem on Primes in Arithmetic Progressions, states that there are infinitely many prime numbers of the form a + bn, where a and b are positive integers that are relatively prime. Peter Gustav Lejeune Dirichlet, a German mathematician, first stated and proved this theorem in 1837. It has applications in other areas of mathematics and can be generalized to other types of progressions. There is a proof for this theorem, but it is quite complex and has been simplified over the years.
  • #1
burritoloco
83
0
Hello,

I'm wondering if this is true, or if anyone has seen this before:

Let q, t be coprime integers. Then there exist infinitely many primes r such that
1. q is primitive root modulo r and
2. r = q + kt, for some k > 0.If we take away 1, this becomes Dirichlet's Thm.

http://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions

But could this be true when we allow 1 ?
 
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  • #2
The problem is definitely tougher than Dirichlet's Thm .!
 
  • #3
Haha, was just wondering if this had been done. It's definitely not in my analytic number theory book!
 
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FAQ: Dirichlet's Theorem on Arithmetic Progressions

What is Dirichlet's Theorem on Arithmetic Progressions?

Dirichlet's Theorem on Arithmetic Progressions, also known as Dirichlet's Theorem on Primes in Arithmetic Progressions, is a mathematical theorem that states that for any given positive integers a and b that are relatively prime, there are infinitely many prime numbers of the form a + bn, where n is a non-negative integer. In other words, there are infinitely many primes that are in the same arithmetic progression.

Who is Dirichlet and what is his contribution to this theorem?

Peter Gustav Lejeune Dirichlet was a German mathematician who first stated and proved this theorem in 1837. He was also one of the founders of modern number theory and made significant contributions to the study of Fourier series, elliptic functions, and the theory of partitions.

What is the significance of Dirichlet's Theorem on Arithmetic Progressions?

This theorem is significant because it provides a way to prove that there are infinitely many prime numbers in certain arithmetic progressions. It also has applications in other areas of mathematics, such as the study of elliptic curves and cryptography.

Can Dirichlet's Theorem be generalized to other types of progressions?

Yes, Dirichlet's Theorem can be generalized to other types of progressions, such as geometric progressions. In fact, there is a more general version of the theorem, known as the Dirichlet's Theorem on Diophantine Approximation, that deals with more general types of progressions.

Is there a proof for Dirichlet's Theorem on Arithmetic Progressions?

Yes, there is a proof for Dirichlet's Theorem on Arithmetic Progressions, but it is quite complex and requires a deep understanding of number theory and analysis. The original proof by Dirichlet was quite long and complicated, but more elegant and simpler proofs have been developed over the years.

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