Gauss’s lemma in number theory
E29548
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Euler’s criterion | 1 |
| Gauss’s golden theorem | 1 |
| Gauss’s lemma in number theory canonical | 1 |
| Legendre symbol takes values ±1 for a coprime to p | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T228962 — 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: Gauss’s lemma in number theory Context triple: [Carl Friedrich Gauss, hasConceptNamedAfter, Gauss’s lemma in number theory]
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A.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
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B.
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.
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C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
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D.
Gaussian integers
Gaussian integers are complex numbers whose real and imaginary parts are both integers, forming a lattice in the complex plane with important applications in number theory and algebra.
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E.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
- 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: Gauss’s lemma in number theory Target entity description: 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.
-
A.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
-
B.
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.
-
C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
D.
Gaussian integers
Gaussian integers are complex numbers whose real and imaginary parts are both integers, forming a lattice in the complex plane with important applications in number theory and algebra.
-
E.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
lemma in number theory
ⓘ
result in elementary number theory ⓘ |
| appearsIn |
Disquisitiones Arithmeticae
ⓘ
surface form:
Gauss’s Disquisitiones Arithmeticae
|
| appliesTo | odd prime modulus ⓘ |
| assumes |
a is an integer coprime to p
ⓘ
p is an odd prime ⓘ |
| category |
results about quadratic residues
ⓘ
theorems about primes ⓘ |
| characterizes |
quadratic non-residues via parity of negative multiples
ⓘ
quadratic residues via parity of negative multiples ⓘ |
| concerns |
quadratic non-residues modulo an odd prime
ⓘ
quadratic residues modulo an odd prime ⓘ |
| domain | modular arithmetic ⓘ |
| equates | Legendre symbol (a|p) with (−1)^n ⓘ |
| equivalentTo | certain combinatorial formulations of the Legendre symbol ⓘ |
| excludes | case where p divides a ⓘ |
| field | number theory ⓘ |
| gives | criterion for quadratic residuosity modulo an odd prime ⓘ |
| hasNotation | (a|p) for the Legendre symbol in its statement ⓘ |
| historicalPeriod | early 19th century ⓘ |
| holdsFor | integers a with gcd(a,p)=1 ⓘ |
| implies |
Gauss’s lemma in number theory
self-linksurface differs
ⓘ
surface form:
Legendre symbol takes values ±1 for a coprime to p
|
| involves |
counting negative representatives among reduced multiples
ⓘ
sequence of multiples a,2a,…,((p−1)/2)a modulo p ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| oftenFormulatedWith | interval (−p/2,p/2] for representatives modulo p ⓘ |
| provides | practical method to compute Legendre symbols ⓘ |
| relatedTo |
Gauss’s lemma in number theory
self-linksurface differs
ⓘ
surface form:
Euler’s criterion
quadratic reciprocity law ⓘ |
| relates |
Legendre symbol
ⓘ
number of sign changes in a sequence of multiples ⓘ |
| requires | reduction of integers modulo p into a symmetric interval around 0 ⓘ |
| statement | Let p be an odd prime and a an integer coprime to p. Consider the numbers a,2a,3a,…,((p−1)/2)a reduced modulo p into the interval (−p/2,p/2]. Let n be the number of these residues that are negative. Then (a|p) = (−1)^n, where (a|p) is the Legendre symbol. ⓘ |
| subfield |
elementary number theory
ⓘ
quadratic number theory ⓘ |
| toolIn | computational number theory ⓘ |
| topic |
Legendre symbol
ⓘ
quadratic residues ⓘ |
| typeOf | criterion for quadratic residuosity ⓘ |
| usedBy | elementary number theory textbooks ⓘ |
| usedFor |
computing Legendre symbols
ⓘ
testing whether an integer is a quadratic residue modulo an odd prime ⓘ |
| usedIn |
elementary proofs of properties of Legendre symbols
ⓘ
proofs of quadratic reciprocity ⓘ |
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: Gauss’s lemma in number theory Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
Gauss’s lemma in number theory
→
relatedTo
→
Gauss’s lemma in number theory
self-linksurface differs
ⓘ
subject surface form:
Gauss’s lemma (number theory)
this entity surface form:
Euler’s criterion
Gauss’s lemma in number theory
→
implies
→
Gauss’s lemma in number theory
self-linksurface differs
ⓘ
subject surface form:
Gauss’s lemma (number theory)
this entity surface form:
Legendre symbol takes values ±1 for a coprime to p
this entity surface form:
Gauss’s golden theorem