Quadratic residue

In modular arithmetic, a quadratic residue for the modulus N is a number which can be expressed as the residue of a² modulo N for some integer a. A quadratic non-residue of N is a number which is not a quadratic residue of N.

Contents 1 Legendre symbol 2 Jacobi symbol 3 See also 4 References

[edit] Legendre symbol

When the modulus is a prime p, the Legendre symbol expresses the quadratic nature of a modulo p. We write

if p divides a;

if a is a quadratic residue of p;

if a is a quadratic non-residue of p.

The Legendre symbol is multiplicative, that is,

[edit] Jacobi symbol

For an odd positive n, the Jacobi symbol is defined as a product of Legendre symbols

where the prime factorisation of n is

The Jacobi symbol is bimultiplicative, that is,

and

If a is a quadratic residue of n then the Jacobi symbol , but the converse does not hold. For example,

but since the Legendre symbol , it follows that 3 is a quadratic non-residue of 5 and hence of 35.

[edit] See also

• Quadratic reciprocity

[edit] References

• G. H. Hardy; E. M. Wright (2008). An Introduction to the Theory of Numbers, 6th ed. Oxford University Press. ISBN 0-19-921986-9. 

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