Definition: Jacobi Symbol

Let $n > 0$ be an odd and positive integer with the factorization $n=p_1^{e_1}\cdots p_r^{e_r}.$ For an integer $a\in\mathbb Z,$ the Jacobi symbol of $a$ modulo $n$ is an arithmetic function defined by

$$\left(\frac an\right):=\left(\frac a{p_1}\right)^{e_1}\cdots \left(\frac a{p_r}\right)^{e_r},$$ where $\left(\frac a{p_i}\right)$ denote the Legendre symbols of $a$ modulo the prime numbers $p_i$ dividing $n.$

Algorithms: 1
Explanations: 2
Proofs: 3
Theorems: 4

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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