Legendre symbol
E171225
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Legendre symbol canonical | 13 |
| Legendre symbol (2/p) | 1 |
| Legendre symbol is only defined for odd prime moduli | 1 |
| the Legendre symbol takes values in {−1,1} for a not divisible by p | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1489704 — 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: Legendre symbol Context triple: [Gauss’s lemma (number theory), topic, Legendre symbol]
-
A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
B.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
-
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.
Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
-
E.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
- 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: Legendre symbol Target entity description: 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.
-
A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
B.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
-
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.
Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
-
E.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
multiplicative character modulo p
ⓘ
number-theoretic function ⓘ quadratic character ⓘ |
| appearsIn |
algebraic number theory
ⓘ
analytic number theory ⓘ elementary number theory ⓘ |
| category | multiplicative arithmetic function ⓘ |
| codomain | {-1,0,1} ⓘ |
| computableBy |
Euler criterion
ⓘ
quadratic reciprocity and supplementary laws ⓘ |
| definedFor | odd prime moduli ⓘ |
| domain | integers ⓘ |
| EulerCriterion | (a/p) ≡ a^{(p-1)/2} (mod p) ⓘ |
| extension | generalized to composite moduli via the Jacobi symbol ⓘ |
| generalization | Jacobi symbol ⓘ |
| historicalOrigin | introduced in the work of Adrien-Marie Legendre on quadratic reciprocity ⓘ |
| multiplicativeProperty |
(a^2/p)=1 if p does not divide a
ⓘ
(ab/p)=(a/p)(b/p) ⓘ |
| namedAfter | Adrien-Marie Legendre ⓘ |
| notation | (a/p) ⓘ |
| orthogonalityProperty | sum_{a mod p} (a/p)=0 ⓘ |
| periodicity | (a/p) depends only on a mod p ⓘ |
| property |
(-1/p)=-1 if p ≡ 3 (mod 4)
ⓘ
(-1/p)=1 if p ≡ 1 (mod 4) ⓘ (0/p)=0 ⓘ (1/p)=1 for any odd prime p ⓘ (2/p)=-1 if p ≡ ±3 (mod 8) ⓘ (2/p)=1 if p ≡ ±1 (mod 8) ⓘ (a/p)=(b/p) if a ≡ b (mod p) ⓘ (a/p)=0 or 1 if a is a square modulo p ⓘ sum_{a=1}^{p-1} (a/p)=0 for odd prime p ⓘ |
| quadraticReciprocity | (p/q)(q/p)=(-1)^{(p-1)(q-1)/4} for odd primes p,q ⓘ |
| relatedConcept |
Dirichlet character modulo p
ⓘ
linked to:
Dirichlet characters
Gauss sum ⓘ quadratic non-residue modulo p ⓘ quadratic residue modulo p ⓘ |
| supplementaryLaw |
(-1/p)=(-1)^{(p-1)/2}
ⓘ
(2/p)=(-1)^{(p^2-1)/8} ⓘ |
| symmetryProperty | (a/p)=(a+kp/p) for any integer k ⓘ |
| usedIn |
Dirichlet L-functions
ⓘ
Gauss sums ⓘ construction of quadratic fields ⓘ primality testing ⓘ quadratic reciprocity ⓘ quadratic residue theory ⓘ |
| valueCondition |
(a/p)=-1 if a is a quadratic non-residue modulo p
ⓘ
(a/p)=0 if p divides a ⓘ (a/p)=1 if a is a quadratic residue modulo p and a not congruent 0 mod p ⓘ |
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: Legendre symbol Description of subject: 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.
Referenced by (16)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Gauss’s lemma in number theory
subject linked to:
Gauss’s lemma in number theory
linked to: Legendre symbol
Euler criterion
→
hasConsequence
→
the Legendre symbol takes values in {−1,1} for a not divisible by p
ⓘ
linked to: Legendre symbol
linked to: Legendre symbol
subject linked to:
Legendre