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.
 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,
 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,
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.
 See also
- 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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