Editing Quadratic residue (section)

===Least quadratic non-residue===
The least quadratic residue mod ''p'' is clearly 1.  The question of the magnitude of the least quadratic non-residue ''n''(''p'') is more subtle, but it is always prime, with 7 appearing for the first time at 71.

The Pólya–Vinogradov inequality above gives O({{radic|''p''}} log ''p''). 

The best unconditional estimate is ''n''(''p'') ≪ ''p''<sup>θ</sup> for any θ>1/4{{radic|e}}, obtained by estimates of Burgess on [[character sum]]s.<ref name=FI156/> 

Assuming the [[Generalised Riemann hypothesis]], Ankeny obtained ''n''(''p'') ≪ (log ''p'')<sup>2</sup>.<ref>{{cite book | title=Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis | first=Hugh L. | last=Montgomery | author-link=Hugh Montgomery (mathematician) | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 | page=176 }}</ref> 

[[Linnik]] showed that the number of ''p'' less than ''X'' such that ''n''(''p'') > X<sup>ε</sup> is bounded by a constant depending on ε.<ref name=FI156>{{cite book | title=Opera De Cribro | first1=John B. | last1=Friedlander | author1-link=John Friedlander | first2=Henryk  | last2=Iwaniec | author2-link=Henryk Iwaniec | publisher=[[American Mathematical Society]] | year=2010 | isbn=978-0-8218-4970-5 | zbl=1226.11099  | page=156 }}</ref>

The least quadratic non-residues mod ''p'' for odd primes ''p'' are:
:2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, ... {{OEIS|id=A053760}}