Quadratic Residue


Quadratic Residue
a prime number except 2
If has an integer solution, is a quadratic residue of divisor and a quadratic nonresidue otherwise.

Consider .
is a quadratic residue
  
is a quadratic nonresidue
  

Legendre's symbols
  1. , is a quadratic residue of
  2. , is a quadratic nonresidue of


If , then

Euler's criterion

First supplementary law

Second supplementary law

Reciprocity law
a prime number other than 2


Jacobi's symbols
an odd number other than 1
(factorization into prime numbers)
If , then


  1. an odd number other than 1

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