Properties of Legendre Symbol
1. Properties of Legendre Symbol
- Multiplicativity \[(a/p)(b/p) \rightarrow (ab/p)\]
- Euler's Criterion \[(a/p)=a^{(p-1)/2}\]
- Modular Equivalence \[a\equiv b \pmod{p} \rightarrow (a/p)=(b/p)\]
- Quadratic Reciprocity \[(p/q)(q/p) = (-1)^{\frac{p-1}{2}\cdot\frac{q-1}{2}}\]
- \(a = -1\) \[(-1/p) = (-1)^{\frac{p-1}{2}}\]
- \(a = 2\) \[(2/p) = (-1)^{\frac{p^2-1}{2}}\]
- \(a = x^2\) \[p\nmid x \leftrightarrow (x^2/p) = 1\]