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

How do you determine if a number is 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 prove a quadratic residue?

If is even, say k = 2 m , then a ( p − 1 ) / 2 = g ( p − 1 ) m = 1 . 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 .

What is a quadratic residue mod p?

From Prime-Wiki. 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 .

How do you find quadratic residue and non residue?

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: If p ≡ 1 (mod 4) the negative of a residue modulo p is a residue and the negative of a nonresidue is a nonresidue.

What is a quadratic non residue?

If there is no integer such that. i.e., if the congruence (35) has no solution, then is said to be a quadratic nonresidue (mod ). If the congruence (35) does have a solution, then is said to be a quadratic residue (mod ).

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

We say that a ∈ Z is a quadratic residue mod n if there exists b ∈ Z such that a ≡ b2 mod n. If a, b are quadratic residues mod n then so is ab. 10.2 Prime moduli. We are mainly interested in quadratic residues modulo a prime.

Is a quadratic residue mod 7?

Thus 1,2,4 are quadratic residues modulo 7 while 3,5,6 are quadratic nonresidues modulo 7. has exactly two incongruent solutions modulo p.

Is 2 a quadratic residue mod p?

2(p-1)/2 ≡ (−1)2k+2 ≡ 1 (mod p), so Euler’s Criterion tells us that 2 is a quadratic residue. This proves that 2 is a quadratic residue for any prime p that is congruent to 7 modulo 8.

How to determine the number of solutions of a quadratic formula?

1. How to determine the number of solutions. The key is the expression in the square root: . In general there are three cases: is positive. The square root of a positive number is also some positive number. So in the numerator of the quadratic formula we will get two values: (-b + the square root) and (-b – the square root).

How many real number roots does a quadratic equation have?

if , then the quadratic has two distinct real number roots. Furthermore, if is a perfect square number, then the roots will be rational, otherwise the roots of the equation will be a conjugate pair of irrational numbers of the form where if , then the quadratic has a single real number root with a multiplicity of 2.

How do you determine the character of a quadratic equation?

The discriminant can be evaluated to determine the character of the solutions of a quadratic equation, thus: if , then the quadratic has two distinct real number roots. if , then the quadratic has a single real number root with a multiplicity of 2.

What is the multiplicity of the root of a quadratic equation?

if , then the quadratic has a single real number root with a multiplicity of 2. In this case the quadratic is a perfect square having two factors: , hence is the root, and the multiplicity of 2 comes from the fact that there are two identical factors. if , then the quadratic has no real number solutions.