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NC Ankeny showed assuming Riemann Hypothesis the least quadratic non residue( let it be '$r$') modulo some prime $p$ to be $O(\log^2 p)$. It is easy to see that $r$ is a prime.

I have following questions

  1. Can we determine for which prime $p$ the least quadratic non-residue(lqnr) is $3 \bmod 4$.
  2. (This is a weaker question) Can we find any prime non residue which is congruent to $3 \bmod 4$
  3. Can we lower bound number of prime non-residue.(Anything better than constant would be interesting).

Note that everywhere I want non-residue $< p$.

Some experimental results are :

for $p=1\bmod 16$ the probability(experimental) of lqnr is $172/241$.

for $p=1\bmod 32$ the probability(experimental) of lqnr is $75/110.$

for $p=1\bmod 64$ the probability(experimental) of lqnr is $107/156.$

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2) Of course we can. Choose any non-residue $a$ and consider any prime congruent to $a$ modulo $p$ and to 3 modulo 4, such prime exists by Dirichlet theorem on primes in arithmetic progressions (with difference $4p$ in our case.)

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  • $\begingroup$ I want non-residue < p. $\endgroup$
    – xyz
    Commented Jul 16, 2016 at 14:28
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    $\begingroup$ @xyz Please specify this in your post, it is not clear at all. $\endgroup$
    – Fedor Petrov
    Commented Jul 16, 2016 at 14:30

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