The least quadratic non-residue

Question

For my thesis I need a good bound for the least quadratic non-residue modulo an odd prime $p$, which I can cite as proven. So I researched a lot and found several papers and bound. As far as I read in Wikipedia, there is the bound $$ p^{\frac{1}{4}\sqrt{\mathrm{e}}}. $$ This is cited by an article, but when I read the article, it says this is >>the best known estimate<<. So I get the feeling that this is not proven. Or do I understand this wrong?

Answer 1

For p = 3 or 5 (mod 8) the lowest non quadratic residue is 2. So you are looking for results where p = 1 or 7 (mod 8). This is a result of Gauss in the Disquisitiones. Theorem 5.2 you say is all you need, but p = 3 gives too low a bound. 2 is the lowest non residue possible. Thus divide primes into two segments given above and this should solve the problem.