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(c) 2024, Andrew N. Fargo

Properties of Legendre Symbol

1. Properties of Legendre Symbol

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

Date: 2024-11-18 Mon 13:36

Author: Andrew N. Fargo

Created: 2024-11-25 Mon 16:06

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