representants of quadratic residues

Theorem.  Let p be a positive odd prime number.  Then the integers

12, 22,,(p-12)2 (1)

constitute a representant system of incongruent quadratic residuesMathworldPlanetmath modulo p.  Accordingly, there are p-12 quadratic residues and equally many nonresidues modulo p.

Proof.  Firstly, the numbers (1), being squares, are quadratic residues modulo p.  Secondly, they are incongruent, because a congruenceMathworldPlanetmatha2b2(modp)  would imply


which is impossible when a and b are different integers among 1, 2,,p-12.  Third, if c is any quadratic residue modulo p, and therefore the congruence  x2c(modp)  has a solution x, then x is congruent with one of the numbers


which form a reduced residue systemMathworldPlanetmath modulo p (see absolutely least remainders).  Then x2 and c are congruent with one of the numbers (1).

Title representants of quadratic residues
Canonical name RepresentantsOfQuadraticResidues
Date of creation 2013-03-22 19:00:35
Last modified on 2013-03-22 19:00:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 11A15
Synonym representant system of quadratic residues
Related topic GaussianSum
Related topic DifferenceOfSquares
Related topic DivisibilityByPrimeNumber