I heard that if $p$ is an odd prime, there is a quadratic nonresidue modulo $p$ less than $p^{1/2} +1$. How to show this statement?
bound of quadratic nonresidue modulo an odd prime
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number-theory
quadratic-reciprocity
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0This http://oeis.org/A000229 is, roughly speaking, a list of primes with astonishingly large first quadratic nonresidues. And the first nonresidues are still tiny. So, computational experience says pretty strongly that one can do better than $\sqrt p,$ but proofs are hard to come by. – 2012-05-28