Quadratic Residues and the Quadratic Reciprocity Law

  • Tom M. Apostol
Part of the Undergraduate Texts in Mathematics book series (UTM)


As shown in Chapter 5, the problem of solving a polynomial congruence
$$ f\left( x \right) \equiv 0\left( {\bmod m} \right) $$
can be reduced to polynomial congruences with prime moduli plus a set of linear congruences. This chapter is concerned with quadratic congruences of the form
$$ {x^2} \equiv n\;(\,\bmod \,\;p) $$
where p is an odd prime and n ≢ 0 (mod p). Since the modulus is prime we know that (1) has at most two solutions. Moreover, if x is a solution so is - x, hence the number of solutions is either 0 or 2.


Diophantine Equation Quadratic Residue Legendre Symbol Linear Congruence Jacobi Symbol 
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Copyright information

© Springer Science+Business Media New York 1976

Authors and Affiliations

  • Tom M. Apostol
    • 1
  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

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