The legendre(a, p) function computes the Legendre symbol  of a and p, which is defined to be  if a is a quadratic residue ,  if a is a quadratic non-residue , and  if a is congruent to . The number a is a quadratic residue mod p if it is not a multiple of p and has a square root , that is, there is an integer  such that  is congruent to . The number a is a quadratic non-residue mod p if it is not a multiple of p and does not have a square root .

Note: The legendre routine returns unevaluated if the given algebraic arguments are not of the types specified above.

The command with(numtheory,legendre) allows the use of the abbreviated form of this command.