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 $107/156.$

• You should include the stackexchange link: math.stackexchange.com/questions/1861216/… Jul 16, 2016 at 13:12
• @NickS How will it benefit the question ? Or you are saying put question on only one platform.
– xyz
Jul 16, 2016 at 15:18
• Jul 16, 2016 at 15:24
• @NickS thanks I have deleted it from there.
– xyz
Jul 16, 2016 at 15:28
• For problem 2, see Theorem 1 of pollack.uga.edu/gica4.pdf Feb 27, 2018 at 5:10

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.)