Calculate The Values of The Following Legendre Symbols

What is a Legendre Symbol?

The Legendre symbol, denoted by (a/p), is a mathematical function that provides information about the solvability of the congruence x² ≡ a mod p, where p is an odd prime number and a is an integer not divisible by p.

The Legendre symbol can take three possible values:

• 1 if there exists an integer x such that x² ≡ a mod p
• -1 if no such integer exists
• 0 if a is divisible by p

How to Calculate Legendre Symbols

Calculating Legendre symbols involves several steps that can be performed using quadratic reciprocity and other number-theoretic properties. Here's a step-by-step method:

1. First, ensure that p is an odd prime and that a is not divisible by p.
2. If a is congruent to 1 modulo p, then (a/p) = 1.
3. If a is congruent to -1 modulo p, then (a/p) = (-1)^((p-1)/2).
4. For other values of a, use quadratic reciprocity to express (a/p) in terms of (p/a).
5. Continue applying quadratic reciprocity until you reach a base case where a is 1 or -1.

Let's look at an example to illustrate this process.

Example Calculation

Calculate (7/19):

1. 19 is an odd prime and 7 is not divisible by 19.
2. Apply quadratic reciprocity: (7/19) = (-1)^((7-1)(19-1)/4) (19/7).
3. Calculate the exponent: (6×18)/4 = 27, so (-1)^27 = -1.
4. Now calculate (19/7). Since 19 ≡ 5 mod 7, we have (19/7) = (5/7).
5. Again apply quadratic reciprocity: (5/7) = (-1)^((5-1)(7-1)/4) (7/5) = (-1)^(6) (7/5) = (7/5).
6. Now calculate (7/5). Since 7 ≡ 2 mod 5, we have (7/5) = (2/5).
7. Since 2 is a quadratic residue modulo 5, (2/5) = 1.
8. Working backwards: (7/5) = 1, (5/7) = 1, (19/7) = 1, so (7/19) = -1 × 1 = -1.

Key Properties of Legendre Symbols

Legendre symbols have several important properties that make them useful in number theory:

1. Multiplicativity: (ab/p) = (a/p)(b/p) for any integers a and b.
2. Quadratic Reciprocity: For two distinct odd primes p and q, (p/q) = (-1)^((p-1)(q-1)/4) (q/p).
3. Complementary Property: (a/p) = (-1)^((a-1)(p-1)/4) (p/a) for a odd and not divisible by p.
4. Gauss's Lemma: (a/p) = (-1)^s, where s is the number of least positive residues of a, 2a, 3a, ..., ((p-1)/2)a that are greater than p/2.

These properties allow for efficient computation of Legendre symbols and provide deep insights into the distribution of quadratic residues.

Applications of Legendre Symbols

Legendre symbols have numerous applications in number theory and related fields:

• Solving quadratic congruences: They help determine whether a quadratic congruence has solutions.
• Constructing quadratic fields: They are used in the construction of quadratic number fields.
• Cryptography: They play a role in some cryptographic algorithms and protocols.
• Number theory research: They are fundamental in many advanced number-theoretic results.

Understanding Legendre symbols provides a foundation for more advanced topics in number theory and its applications.