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单词 RepresentantsOfQuadraticResidues
释义

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

pa+borpa-b,

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

±1,±2,,±p-12

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

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更新时间:2025/5/4 4:46:15