The Legendre symbol \(\left( \frac{a}{p} \right)\) is a notation that indicates whether an integer \(a\) is a quadratic residue modulo an odd prime \(p\). It is defined as follows:

\[
\left( \frac{a}{p} \right) = 
\begin{cases} 
0 & \text{if } a \equiv 0 \pmod{p}, \\
1 & \text{if } a \not\equiv 0 \pmod{p} \text{ and there exists an integer } x \text{ such that } x^2 \equiv a \pmod{p}, \\
-1 & \text{if } a \not\equiv 0 \pmod{p} \text{ and there does not exist an integer } x \text{ such that } x^2 \equiv a \pmod{p}.
\end{cases}
\]

For the Legendre symbol \(\left( \frac{2029}{3} \right)\), we need to determine whether 2029 is a quadratic residue modulo 3.

### Step-by-Step Calculation:

1. **Reduce 2029 modulo 3:**

   We first find \(2029 \mod 3\). To do this, we sum the digits of 2029 and check if this sum is divisible by 3. 

   \[
   2029 = 2 + 0 + 2 + 9 = 13
   \]

   Now, sum the digits of 13:

   \[
   13 = 1 + 3 = 4
   \]

   Since 4 is not divisible by 3, we reduce \(2029 \mod 3\) by dividing 2029 by 3 and finding the remainder.

   \[
   2029 \div 3 = 676 \quad \text{remainder } 1
   \]

   Therefore, \(2029 \equiv 1 \pmod{3}\).

2. **Apply the properties of the Legendre symbol:**

   Since \(2029 \equiv 1 \pmod{3}\), we have:

   \[
   \left( \frac{2029}{3} \right) = \left( \frac{1}{3} \right)
   \]

   The Legendre symbol \(\left( \frac{1}{p} \right) = 1\) for any odd prime \(p\) because 1 is always a quadratic residue modulo any prime.

### Conclusion:

\[
\left( \frac{2029}{3} \right) = \boxed{1}
\]

This means that 2029 is a quadratic residue modulo 3.

Would you like more details, or do you have any other questions?

Here are 5 additional related questions you might find interesting:
1. How do we compute the Legendre symbol for larger numbers?
2. What are the rules for determining if a number is a quadratic residue modulo a prime?
3. How is the Legendre symbol related to quadratic reciprocity?
4. How do you compute the Jacobi symbol, a generalization of the Legendre symbol?
5. What are the applications of the Legendre and Jacobi symbols in number theory?

**Tip:** When computing the Legendre symbol \(\left( \frac{a}{p} \right)\), reduce \(a\) modulo \(p\) first to simplify your calculations.

Explore the Legendre symbol and its application in determining quadratic residues modulo a prime number. Detailed solution and step-by-step calculation for the Legendre symbol (2029/3).

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