if P is an odd prime then ∑_(n=0)^(p-1)▒(n/p) =0

To prove that \(\sum_{n=0}^{p-1} \left(\frac{n}{p}\right) = 0\), where \(p\) is an odd prime and \(\left(\frac{n}{p}\right)\) denotes the Legendre symbol, follow these steps:

### Legendre Symbol:
The Legendre symbol \(\left(\frac{n}{p}\right)\) is defined as:
\[
\left(\frac{n}{p}\right) =
\begin{cases} 
1 & \text{if } n \text{ is a quadratic residue modulo } p, \\
-1 & \text{if } n \text{ is a quadratic non-residue modulo } p, \\
0 & \text{if } n \equiv 0 \pmod{p}.
\end{cases}
\]

### Sum of Legendre Symbols:
We need to evaluate the sum:
\[
\sum_{n=0}^{p-1} \left(\frac{n}{p}\right).
\]

This sum can be split as:
\[
\sum_{n=0}^{p-1} \left(\frac{n}{p}\right) = \left(\frac{0}{p}\right) + \sum_{n=1}^{p-1} \left(\frac{n}{p}\right).
\]

Since \(\left(\frac{0}{p}\right) = 0\), we have:
\[
\sum_{n=0}^{p-1} \left(\frac{n}{p}\right) = \sum_{n=1}^{p-1} \left(\frac{n}{p}\right).
\]

### Properties of the Legendre Symbol:
1. **Number of Residues and Non-Residues**: For any odd prime \(p\), exactly half of the integers \(n\) in \(\{1, 2, \dots, p-1\}\) are quadratic residues, and the other half are non-residues. Therefore, there are \((p-1)/2\) values of \(n\) such that \(\left(\frac{n}{p}\right) = 1\), and \((p-1)/2\) values such that \(\left(\frac{n}{p}\right) = -1\).

2. **Sum of Residues and Non-Residues**: Given the symmetry and the equal count of residues and non-residues, the sum \(\sum_{n=1}^{p-1} \left(\frac{n}{p}\right)\) will consist of \((p-1)/2\) terms of 1 and \((p-1)/2\) terms of -1.

Thus:
\[
\sum_{n=1}^{p-1} \left(\frac{n}{p}\right) = \frac{p-1}{2} \cdot 1 + \frac{p-1}{2} \cdot (-1) = \frac{p-1}{2} - \frac{p-1}{2} = 0.
\]

### Conclusion:
Therefore, \(\sum_{n=0}^{p-1} \left(\frac{n}{p}\right) = 0\).

Would you like more details on any part of the proof? Here are some related questions:

1. How is the Legendre symbol defined for composite numbers?
2. What are quadratic residues and non-residues in more detail?
3. How does the symmetry of quadratic residues contribute to the sum being zero?
4. Can this property be generalized to non-prime numbers?
5. What are some applications of the Legendre symbol in number theory?

**Tip**: Understanding the properties of quadratic residues and non-residues is crucial for studying more advanced topics in number theory, like quadratic reciprocity.

Explore the proof that the sum of Legendre symbols for an odd prime is zero. Understand the properties of Legendre symbols, quadratic residues, and non-residues in number theory. Suitable for advanced undergraduate and graduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation