First of all, for which prime numbers is 1 a quadratic residue?

Zero, while a square, is not considered a quadratic residue.

The properties of quadratic residues are widely used in number theory.

We can find quadratic residues or verify them using the above formula.

Depending on whether or not m is a quadratic residue modulo p.

So every quadratic residue (mod p) makes the first factor zero.

These identies cover all the numbers that are not quadratic residues for those bases.

The least quadratic residue mod p is clearly 1.

If then may or may not be a quadratic residue .

For which primes p is 17 a quadratic residue?

A quadratic residue code is a type of cyclic code.

See quadratic residue for more information about extracting square roots in finite fields.

This is indeed correct, because 7 is not a quadratic residue modulo 11.

It concerns the quadratic residue code of order n, extended by adding a single parity check bit.

The quadratic residues form a group under multiplication.

Thus c is a quadratic residue modulo p.

A probabilistic for square root of a quadratic residue modulo a prime.

The construction of these graphs uses quadratic residues.

Modulo a prime, there are an equal number of quadratic residues and nonresidues.

Moreover (for the case that is a quadratic residue, same idea holds for ):

This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, .

Modulo 2, every integer is a quadratic residue.

Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.

Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue.

The code can also be constructed as the quadratic residue code of length 11 over the finite field F.