Jacobi symbol

E182750

generalization of Legendre symbol multiplicative function number-theoretic function

The Jacobi symbol is a number-theoretic function that generalizes the Legendre symbol and plays a key role in quadratic residues and primality testing in modular arithmetic.

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All labels observed (1)

Label	Occurrences
Jacobi symbol canonical	4

How this entity was disambiguated

This entity first appeared as the object of triple T1615215 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Jacobi symbol
Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi symbol]

A. Legendre symbol
The Legendre symbol is a number-theoretic function that indicates whether an integer is a quadratic residue modulo an odd prime, taking values 1, −1, or 0 accordingly.
B. quadratic reciprocity law
The quadratic reciprocity law is a fundamental theorem in number theory that characterizes when a quadratic equation modulo one odd prime has solutions in terms of solvability modulo another, revealing a deep symmetry between primes.
C. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
D. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
E. Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Jacobi symbol
Target entity description: The Jacobi symbol is a number-theoretic function that generalizes the Legendre symbol and plays a key role in quadratic residues and primality testing in modular arithmetic.

A. Legendre symbol
The Legendre symbol is a number-theoretic function that indicates whether an integer is a quadratic residue modulo an odd prime, taking values 1, −1, or 0 accordingly.
B. quadratic reciprocity law
The quadratic reciprocity law is a fundamental theorem in number theory that characterizes when a quadratic equation modulo one odd prime has solutions in terms of solvability modulo another, revealing a deep symmetry between primes.
C. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
D. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
E. Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	generalization of Legendre symbol ⓘ multiplicative function ⓘ number-theoretic function ⓘ
algorithmicProperty	computable in polynomial time in log n ⓘ efficiently computable via Euclidean algorithm ⓘ
appearsIn	algebraic number theory ⓘ computational number theory ⓘ elementary number theory textbooks ⓘ
codomain	{-1,0,1} ⓘ
definedOn	pair of integers (a,n) ⓘ
differenceFrom	Legendre symbol ⓘ surface form: Legendre symbol is only defined for odd prime moduli value 1 does not guarantee a is a quadratic residue modulo composite n ⓘ
domainCondition	n is a positive odd integer ⓘ
field	number theory ⓘ
introducedBy	Carl Gustav Jacob Jacobi ⓘ
notation	(a/n) ⓘ
property	(a/n)=0 if and only if gcd(a,n)≠1 ⓘ (a/n)=1 or -1 when gcd(a,n)=1 ⓘ can be computed without factoring n ⓘ completely multiplicative in the top argument a ⓘ depends only on a modulo n ⓘ generalizes Legendre symbol to composite odd moduli ⓘ is a real-valued character modulo n when gcd(a,n)=1 ⓘ multiplicative in the bottom argument n when n is odd and positive ⓘ periodic in a with period n ⓘ
quadraticReciprocity	(a/n)(n/a)=(-1)^{((a-1)/2)((n-1)/2)} for odd coprime a,n ⓘ
relatedConcept	Dirichlet characters ⓘ surface form: Dirichlet character Hilbert symbol ⓘ quadratic character modulo n ⓘ
relatedTo	Kronecker symbol ⓘ Legendre symbol ⓘ
satisfies	(-1/n)=(-1)^{(n-1)/2} ⓘ (0/n)=0 for all n>1 ⓘ (1/n)=1 for all odd positive n ⓘ (2/n)=(-1)^{(n^2-1)/8} ⓘ (a/mn)=(a/m)(a/n) for odd coprime m,n ⓘ (ab/n)=(a/n)(b/n) ⓘ quadratic reciprocity law ⓘ
usedFor	constructing pseudorandom generators based on quadratic residues ⓘ testing quadratic residuosity modulo odd composite n ⓘ
usedIn	Fermat primality test ⓘ surface form: Solovay–Strassen primality test cryptographic algorithms involving quadratic residues ⓘ modular arithmetic ⓘ primality testing ⓘ probabilistic primality testing algorithms ⓘ quadratic residue computations modulo composite n ⓘ quadratic residue theory ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Carl Gustav Jacob Jacobi → notableWork → Jacobi symbol ⓘ

Legendre symbol → generalization → Jacobi symbol ⓘ

Jacobi → knownFor → Jacobi symbol ⓘ

subject surface form: Carl Gustav Jacob Jacobi

Carl → notableWork → Jacobi symbol ⓘ