Is a quadratic residue modulo p?

Is a quadratic residue modulo p?

In mathematics, a number q is called a quadratic residue modulo p if there exists an integer x such that: x 2 ≡ q ( m o d p ) Otherwise, q is called a quadratic non-residue. In effect, a quadratic residue modulo p is a number that has a square root in modular arithmetic when the modulus is p .

What does it mean for a ∈ Z to be a quadratic residue modulo n?

Quadratic Residues. 10.1 Introduction. Definition 10.1. We say that a ∈ Z is a quadratic residue mod n if there exists b ∈ Z such that a ≡ b2 mod n. If there is no such b we say that a is a quadratic non-residue mod n.

For which primes p is 13 a quadratic residue?

For example when p = 13 we may take g = 2, so g2 = 4 with successive powers 1,4,3,12,9,10 (mod 13). These are the quadratic residues; to get the quadratic nonresidues multiply them by g = 2 to get the odd powers 2,8,6,11,5,7 (mod 13).

Which of the following is the quadratic non-residue of the Prime 11?

The quadratic residues modulo 11 are 1,3,4,5 and 9. The quadratic non-residues modulo 11 are 2,6,7,8 and 10.

For which primes is 5 a quadratic residue?

Law of quadratic reciprocity

How do you find a 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.

How do you tell if a number is a quadratic residue?

IS 31 is a quadratic residue in modulo 67?

Solution: No. We will use quadratic reciprocity. Note that 67 ≡ 31 ≡ 3 mod 4, and 31 and 67 are primes: (31 67 ) = − (67 31 ) = − ( 5 31 ) = − (31 5 ) = − (1 5 ) = −1.

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.

How do you check for quadratic residue?

How do you check if something is a quadratic residue?

In other words, we have proved Euler’s Criterion, which states is a quadratic residue if and only if a ( p − 1 ) / 2 = 1 , and is a quadratic nonresidue if and only if a ( p − 1 ) / 2 = − 1 . Example: We have is a quadratic residue in if and only if p = 1 ( mod 4 ) .

Which is prime modulus has a quadratic residue?

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.

Which is the product of two nonresidues modulo a prime?

Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue. The first supplement to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then −1 is a nonresidue modulo p. This implies the following:

How many residues are there in a quadratic residue?

Quadratic Residues x 1 2 3 4 x2 1 4 9 16 mod 1 0 0 0 0 mod 2 1 0 1 0 mod 3 1 1 0 1

Is the product of two nonresidues a residue or a nonzero?

Following this convention, modulo an odd prime number there are an equal number of residues and nonresidues. Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue.