any nonzero integer is quadratic residue
Theorem. For every nonzero integer there exists an odd prime number such that is a quadratic residue modulo .
Proof. . We see that and , whence 2 is a quadratic residue modulo .
but . The number (which is odd and ) has an odd prime factor which does not divide . Thus is a quadratic residue modulo .
. We state that and . Therefore 3 is a quadratic residue modulo 13.
. We see that and , i.e. 5 is a quadratic residue modulo 11.
but , . Now the number (which is odd and ) has an odd prime factor . Moreover, since . Accordingly, is a quadratic residue modulo .