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See all on WikipediaJacobi symbol - Wikipedia
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are … See more
For any integer a and any positive odd integer n, the Jacobi symbol (a/n) is defined as the product of the Legendre symbols corresponding to the prime factors of n: See more
The above formulas lead to an efficient O(log a log b) algorithm for calculating the Jacobi symbol, analogous to the Euclidean algorithm for finding the gcd of two numbers. (This should not be surprising in light of rule 2.)
1. See moreThere is another way the Jacobi and Legendre symbols differ. If the Euler's criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol, and in fact may not even be −1 or 1. For example, See more
The following facts, even the reciprocity laws, are straightforward deductions from the definition of the Jacobi symbol and the corresponding properties of the Legendre symbol. See more
The Legendre symbol (a/p) is only defined for odd primes p. It obeys the same rules as the Jacobi symbol (i.e., reciprocity and the supplementary formulas for (−1/p) and (2/p) and multiplicativity of the "numerator".)
Problem: Given … See more• Kronecker symbol, a generalization of the Jacobi symbol to all integers.
• Power residue symbol, a generalization of the Jacobi symbol to higher powers residues. See more• Calculate Jacobi symbol Archived 2016-10-05 at the Wayback Machine shows the steps of the calculation. See more
Wikipedia text under CC-BY-SA license Jacobi Symbol -- from Wolfram MathWorld
Mar 5, 2025 · The Jacobi symbol is a generalization of the Legendre symbol for odd integers. Learn how to compute it, its relation to quadratic residues and the quadratic reciprocity law, and how to use it in the Wolfram Language.
Jacobi Symbol | Brilliant Math & Science Wiki
Learn about the Jacobi symbol, a generalization of the Legendre symbol that can simplify computations involving quadratic residues. Find out how to use it to state and prove an extended version of the law of quadratic reciprocity and to test …
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5.7: Jacobi Symbol - Mathematics LibreTexts
Jul 7, 2021 · In this section, we define the Jacobi symbol which is a generalization of the Legendre symbol. The Legendre symbol was defined in terms of primes, while Jacobi symbol will be …
Learn how to define and use the Jacobi symbol, a generalization of the Legendre symbol for odd moduli. See the main properties of the Jacobi symbol, such as multiplicativity, reciprocity and …
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The Jacobi Symbol - Millersville University of …
You can extend the definition to allow an odd positive number on the bottom using the Jacobi symbol. Most of the properties of Legendre symbols go through for Jacobi symbols, which makes Jacobi symbols very convenient for …
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Learn the definitions, properties and applications of the Legendre and Jacobi symbols, which are used to determine quadratic residues and non-residues modulo an odd integer. See examples, …
elementary number theory - Proofs of the properties of Jacobi …
As @tc1729 notes, these are mostly immediate consequences of the definition of the Jacobi Symbol and the corresponding properties of the Legendre symbol. Proofs may be easily found …
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Jacobi symbols
Learn how to define and use Jacobi symbols, which are a generalization of Legendre symbols for odd moduli. See examples of how they can be applied to Diophantine equations and quadratic …
Jacobi Symbol | SpringerLink
Jan 1, 2025 · The Jacobi symbol was introduced by C.G.J. Jacobi in 1837. The Jacobi symbol of an integer x modulo an odd positive integer n is the product of the Legendre symbols of x …
The Jacobi symbol remains useful for calculating Legendre symbols, because it satis es the same reciprocity and simplifying relations as the Legendre sym-bol (as we now demonstrate), and at …
Learn how to extend the Legendre symbol to the Jacobi symbol for odd moduli. See how to use the Jacobi symbol to test quadratic residues and characterise squares modulo primes.
Quadratic reciprocity, Legendre symbols, and Jacobi symbols
Jun 30, 2018 · The Jacobi symbol is a generalization of the Legendre symbol, using the same notation but allowing the “denominator” to be any odd positive number. As before the symbol …
Jacobi symbol - Encyclopedia of Mathematics
Nov 11, 2023 · The Jacobi symbol is a generalization of the Legendre symbol and has similar properties. In particular, the reciprocity law: $$\left (\frac PQ\right)\left (\frac QP\right)= (-1)^ { …
Computing the Jacobi symbol - Mathematics Stack Exchange
I have been asked to compute $\left (\frac {77} {257}\right)$ specifically using Jacobi symbols, showing all working. I have done the following: $\left (\frac {77} {257}\right) =\left (\frac {257} …
Jacobi symbol - OeisWiki
The Jacobi symbol is a generalization of the Legendre symbol. Given two coprime integers and , with the former the product of primes (not necessarily distinct), [1] the Jacobi symbol is
Definition:Jacobi Symbol - ProofWiki
It can be seen that the Jacobi symbol is a generalization of the Legendre symbol for a composite denominator . In order to determine the quadratic character of an integer modulo a composite …
Then p ≡ a a(p−1)/2 (mod p). The Jacobi symbol extends the domain of the Legendre symbol. Definition: The Jacobi symbol is a function of two integers a and n, written a , that is defined for …
DLMF: §27.9 Quadratic Characters ‣ Multiplicative Number …
The Jacobi symbol (n | P) is a Dirichlet character (mod P). Both (27.9.1) and (27.9.2) are valid with p replaced by P; the reciprocity law (27.9.3) holds if p, q are replaced by any two relatively …
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