Jacobi symbol
E182750
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.
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 is only defined for odd prime moduli
ⓘ
linked to:
Legendre symbol
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 character
ⓘ
linked to:
Dirichlet characters
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 |
Solovay–Strassen primality test
ⓘ
linked to:
Fermat 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jacobi symbol Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.