How do you know if something is quadratic residue?

How do you know if something is quadratic residue?

We only need to solve, when a number (b) has a square root modulo p, to solve quadratic equations modulo p. Given a number a, s.t., gcd(a, p) = 1; a is called a quadratic residue if x2 = a mod p has a solution otherwise it is called a quadratic non-residue.

What do you mean by quadratic residue?

From Wikipedia, the free encyclopedia. In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n.

Is a quadratic residue if and only if?

x2 ≡ a mod m has a solution. has a solution, that is, a is a quadratic residue of p if and only if either p divides a or a(p-1)/2 ≡ 1. If a is not a quadratic residue then a(p-1)/2 ≡ −1.

Is quadratic residue mod p?

Prime modulus Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler’s criterion. (In other words, every congruence class except zero modulo p has a multiplicative inverse.

What is the difference between perfect perfect squares and quadratic residues?

Perfect squares are integers that are the square of an integer, while quadratic residues are elements in the ring of integers modulo some [math]m[/math] which are the square of some other element of this ring. It’s very natural to identify the integers 0, 1, 2, 3, 4, 5, 6 with the residues modulo 7, but they are not the same thing.

What is a quadratic residue modulo m?

If a and m are coprime integers, then a is called a quadratic residue modulo m if the congruence x2 ≡ a (mod m) has a solution. Likewise, if it has no solution, then it is called a quadratic non-residue modulo m .

How do you find the quadratic residue of a given number?

Article 95 introduces the terminology “quadratic residue” and “quadratic nonresidue”, and states that if the context makes it clear, the adjective “quadratic” may be dropped. For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1., n − 1.

What is the least quadratic residue mod p?

The least quadratic residue mod p is clearly 1. The question of the magnitude of the least quadratic non-residue n(p) is more subtle, but it is always prime. The Pólya–Vinogradov inequality above gives O(√p log p).