What is the quadratic reciprocity theorem?

What is the quadratic reciprocity theorem?

Theorem 3.12. 3 (Quadratic Reciprocity Theorem) If p and q are distinct odd primes, then (pq)(qp)=(−1)((p−1)/2)((q−1)/2).

What is quadratic residue and quadratic reciprocity law?

In number theory, the law of quadratic reciprocity is a theorem about quadratic residues modulo an odd prime.

How do you find the quadratic residue of a number?

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 is the purpose of quadratic reciprocity?

The law of quadratic reciprocity is a fundamental result of number theory. Among other things, it provides a way to determine if a congruence x2 ≡ a (mod p) is solvable even if it does not help us find a specific solution.

Why is the law of quadratic reciprocity important?

The law can be used to tell whether any quadratic equation modulo a prime number has a solution. This is important in cryptography and in computer security. Quadratic Reciprocity is a particularly useful tool when you want to see if a number is a square mod p (p prime.)

Who proved quadratic reciprocity?

Gauss
proof of the law of quadratic reciprocity. The law was regarded by Gauss, the greatest mathematician of the day, as the most important general result in number theory since the work of Pierre de Fermat in the 17th century. Gauss also gave the first rigorous proof of the law.

Why is quadratic reciprocity important?

What is the use of quadratic residue?

Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

Is 2 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.

What is the law of quadratic reciprocity for the Legendre symbol?

The law of quadratic reciprocity, noticed by Euler and Legendre and proved by Gauss, helps greatly in the computation of the Legendre symbol. First, we need the following theorem: Theorem: Let \\(p\\) be an odd prime and \\(q\\) be some integer coprime to \\(p\\).

What is the quadratic reciprocity law for −1?

The quadratic reciprocity law is the statement that certain patterns found in the table are true in general. Trivially 1 is a quadratic residue for all primes. The question becomes more interesting for −1. Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47.

Who proved the quadratic reciprocity theorem?

The quadratic reciprocity theorem was conjectured by Euler and Legendre and first proved by Gauss, who referred to it as the “fundamental theorem” in his Disquisitiones Arithmeticae and his papers, writing The fundamental theorem must certainly be regarded as one of the most elegant of its type. (Art. 151)

What is the importance of reciprocity law in mathematics?

Generalizing the reciprocity law to higher powers has been a leading problem in mathematics, and has been crucial to the development of much of the machinery of modern algebra, number theory, and algebraic geometry, culminating in Artin reciprocity, class field theory, and the Langlands program .