properties of the Legendre symbol

Let p be an odd prime and let a be an arbitrary integer. Let (ap) be the Legendre symbolMathworldPlanetmath of a modulo p. Then:


The following properties are satisfied:

  1. 1.

    If abmodp then (ap)=(bp).

  2. 2.

    If a0modp then (a2p)=1.

  3. 3.

    If a0modp and b then (a2bp)=(bp).

  4. 4.



The first three properties are immediate from the definition of the Legendre symbol. Remember that (a/p) is 1 if x2amodp has solutions, the value is -1 if there are no solutions, and equals 0 if a0modp.

The fourth property is a consequence of Euler’s criterion. Indeed,

(ap)a(p-1)/2,(bp)b(p-1)/2,and (abp)(ab)(p-1)/2modp.

It is clear then that (a/p)(b/p)(ab/p)modp. Since the numbers involved are all ±1 or 0, the congruenceMathworldPlanetmathPlanetmathPlanetmath also holds with equality in . ∎


Property (4) is somewhat surprising because, in particular, it says that the productPlanetmathPlanetmath of two quadratic non-residues modulo p is a quadratic residue modulo p, which is not at all obvious.

Title properties of the Legendre symbol
Canonical name PropertiesOfTheLegendreSymbol
Date of creation 2013-03-22 16:17:52
Last modified on 2013-03-22 16:17:52
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Theorem
Classification msc 11-00
Related topic EulersCriterion