Can a quadratic residue be a primitive root?
Since 227 is prime, by theorem 6.7, there is at least one prime root. Moreover, by theorem 6.13, there are exactly φ(φ(227)) primitive roots. = 113, so all quadratic residues can not be primitve roots). It means that there is one quadratic nonresidue which is not a prim- itive root.
Is a quadratic residue?
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 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.
Is 2 always a quadratic residue?
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. 21.1.
What do you mean by primitive roots?
A primitive root of a prime is an integer such that (mod ) has multiplicative order (Ribenboim 1996, p. 22). More generally, if ( and are relatively prime) and is of multiplicative order modulo where is the totient function, then is a primitive root of (Burton 1989, p. 187).
What are primitive roots used for?
When primitive roots exist, it is often very convenient to use them in proofs and explicit constructions; for instance, if p is an odd prime and g is a primitive root mod p, the quadratic residues mod p are precisely the even powers of the primitive root.
How many quadratic residues are there?
For an odd prime p, there are (p+1)/2 quadratic residues (counting zero) and (p-1)/2 non-residues. (The residues come from the numbers 02, 12, 22, , {(p-1)/2}2, these are all different modulo p and clearly list all possible squares modulo p.)…quadratic residue.
| modulus | quadratic residues | quadratic non-residues |
|---|---|---|
| 8 | 0,1,4 | 2,3,5,6,7 |
Why are quadratic residues important?
Quadratic reciprocity is important because it provides a bridge between two apparently distinct branches of mathematics, namely the theory of Galois representations and the theory of automorphic forms. L-functions provide the bridge across the two theories.
Why are primitive roots important?
Primitive roots are generators of cyclic groups. This is very important and there are a lot of open problems concerning them, in particular the Artin’s conjecture for primitive roots, which has an important analogue for elliptic curves.
What is meant by primitive root?