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What is the probability that a randomly chosen number N satisfying $N ≡ 1$ mod 3 is prime?

I know that the "probability" that a randomly chosen m $\in \mathbb{Z}$ is prime is $\frac{1}{ln(m)}$

and that $N = 1 + 3k$ for some $k \in \mathbb{Z}$

But I'm not sure where to proceed from here

Doldrums
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  • There's a version of the Prime Number Theorem for primes in arithmetic progressions. – Angina Seng Apr 03 '20 at 22:10
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    Dirichlet's Theorem tells us that the primes $\pmod n$ are "evenly distributed" between the possible (i.e. prime to $n$) residue classes $\pmod n$, with a sensible definition of "evenly distributed". – lulu Apr 03 '20 at 22:14
  • Usually we assume $m\in \mathbb N$ in such problems. – Peter Apr 05 '20 at 12:12
  • @Peter: I think that's a complete answer; I don't think there's much more to say about this. Perhaps you want to post that as an answer so it can be accepted and the question doesn't remain unanswered? – joriki Apr 05 '20 at 12:39

1 Answers1

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Since we know that the number is not divisible by $3$ , the chance increases by factor $1.5$.

Hence the probability is about $$\frac{1.5}{\ln()}$$ for large .

Peter
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