Many of the properties of the Legendre symbol can be generalized for the Jacobi symbol.

Theorem: Properties of the Jacobi Symbol

Let $n,m$ be odd positive integers and let $a,b$ be any integers. The Jacobi symbol fulfills the following properties:

  1. Jacobi symbol for equal residues: If $a(n)\equiv b(n),$ then $\left(\frac an\right)=\left(\frac bn\right).$
  2. Multiplicativity: $\left(\frac {ab}n\right)=\left(\frac an\right)\cdot \left(\frac bn\right).$
  3. Reciprocal Multiplicativity: $\left(\frac {a}{nm}\right)=\left(\frac an\right)\cdot \left(\frac am\right).$
  4. Reciprocity Law: If $n$ and $m$ are co-prime, then $$\left(\frac mn\right)\left(\frac nm\right)=(-1)^{\frac{n-1}{2}\frac{m-1}{2}}.$$
  5. First supplementary law: $$\left(\frac {-1}n\right)=(-1)^{\frac{n-1}{2}}.$$
  6. First supplementary law: $$\left(\frac {-1}n\right)=(-1)^{\frac{n-1}{2}}.$$

Proofs: 1

Explanations: 1
Proofs: 2
Solutions: 3

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  1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
  2. Blömer, J.: "Lecture Notes Algorithmen in der Zahlentheorie", Goethe University Frankfurt, 1997