Dirichlet's Theorem on Arithmetic Progressions

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The discussion explores the validity of a conjecture related to Dirichlet's Theorem on Arithmetic Progressions, specifically regarding the existence of infinitely many primes r that are both a primitive root modulo q and of the form r = q + kt, where q and t are coprime integers. The poster questions whether this holds true when including the integer 1, suggesting that this scenario presents a greater challenge than Dirichlet's original theorem. They note that this topic is not covered in their analytic number theory literature. The inquiry invites insights or references from others familiar with the subject. The conversation highlights ongoing interest in the complexities of prime distribution in arithmetic progressions.
burritoloco
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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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The problem is definitely tougher than Dirichlet's Thm .!
 
Haha, was just wondering if this had been done. It's definitely not in my analytic number theory book!
 
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