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
- Can we determine for which prime $p$ the least quadratic non-residue(lqnr) is $3 \bmod 4$.
- (This is a weaker question) Can we find any prime non residue which is congruent to $3 \bmod 4$
- 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 $107/156.$