An integer for which the congruence
is solvable. If the above congruence is unsolvable, then is called a quadratic non-residue modulo . Euler's criterion: Let be prime. Then an integer coprime with is a quadratic residue modulo if and only if
and is a quadratic non-residue modulo if and only if
|||I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)|
An amusing unsolved problem is the following: Let be a prime with (). Let be the sum of all quadratic non-residues between 0 and , and the sum of all quadratic residues. It is known that . Give an elementary proof.
|[a1]||G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII|
Quadratic residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_residue&oldid=11386