Math Problem Statement
what is (2029|3) (legendre symbol)?
Solution
The Legendre symbol is a notation that indicates whether an integer is a quadratic residue modulo an odd prime . It is defined as follows:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Number Theory
Modular Arithmetic
Quadratic Residues
Formulas
Legendre Symbol
Theorems
Properties of Quadratic Residues
Suitable Grade Level
Advanced Undergraduate