# Primes among quadratic residues modulo a prime

Given a prime $p$, find all quadratic residues modulo $p$. We randomly select a number $n$ among these quadratic residues modulo $p$. What is the probability of $n$ being a prime as $p$ goes to infinity?

I use A132213 and A000224 to estimate the answer. The answer might be approximately 12%. However, I am wondering if there is a standard proof.

• Given the prime number theorem, you can be sure that the answer is not a fixed value. – Joffan Jan 21 '17 at 16:12
• The number of quadratic residues in $\{1,2,\ldots,p-1\}$ mod $p$ is $(p-1)/2$. The number of these which are primes is $O(\dfrac{p}{\log p})$. So surely the probability as $p$ goes to infinity is just zero? – Aravind Jan 22 '17 at 6:48