Chinese remainder theorem
E530312
The Chinese remainder theorem is a fundamental result in number theory that provides conditions and a method for solving systems of simultaneous congruences with pairwise coprime moduli.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Chinese remainder theorem canonical | 3 |
| Chinese Remainder Theorem | 1 |
| Chinese remainder theorem for ideals | 1 |
| Chinese remainder theorem for rings | 1 |
| uses Chinese remainder theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570568 — 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: Chinese remainder theorem Context triple: [Fermat's little theorem, relatedTo, Chinese remainder theorem]
-
A.
Zhu Shijie
Zhu Shijie was a prominent 13th–14th century Chinese mathematician known for his influential works on algebra, polynomial equations, and early forms of Pascal’s triangle.
-
B.
Chen Jingrun
Chen Jingrun was a Chinese mathematician renowned for his groundbreaking work in number theory, particularly his significant contributions toward proving the Goldbach conjecture.
-
C.
Cishousi
Cishousi is a subway station in Beijing named after the nearby historic Cishou Temple, serving passengers on the city's Line 7.
-
D.
Reflections of China
Reflections of China is a Circle-Vision 360° film attraction at EPCOT that showcases China's landscapes, culture, and history through immersive panoramic visuals.
-
E.
Vista Chinesa
Vista Chinesa is a famous hilltop lookout in Rio de Janeiro offering panoramic views of the city, beaches, and surrounding rainforest.
- 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: Chinese remainder theorem Target entity description: The Chinese remainder theorem is a fundamental result in number theory that provides conditions and a method for solving systems of simultaneous congruences with pairwise coprime moduli.
-
A.
Zhu Shijie
Zhu Shijie was a prominent 13th–14th century Chinese mathematician known for his influential works on algebra, polynomial equations, and early forms of Pascal’s triangle.
-
B.
Chen Jingrun
Chen Jingrun was a Chinese mathematician renowned for his groundbreaking work in number theory, particularly his significant contributions toward proving the Goldbach conjecture.
-
C.
Cishousi
Cishousi is a subway station in Beijing named after the nearby historic Cishou Temple, serving passengers on the city's Line 7.
-
D.
Reflections of China
Reflections of China is a Circle-Vision 360° film attraction at EPCOT that showcases China's landscapes, culture, and history through immersive panoramic visuals.
-
E.
Vista Chinesa
Vista Chinesa is a famous hilltop lookout in Rio de Janeiro offering panoramic views of the city, beaches, and surrounding rainforest.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem ⓘ |
| appearsIn | Sunzi Suanjing ⓘ |
| appliesTo |
modular arithmetic
ⓘ
systems of congruences with pairwise coprime moduli ⓘ |
| associatedWith | Sunzi ⓘ |
| category |
arithmetic theorems
ⓘ
theorems in abstract algebra ⓘ |
| describes | solutions of systems of simultaneous congruences ⓘ |
| ensures | every residue system modulo product corresponds to a tuple of residues modulo each modulus ⓘ |
| field | number theory ⓘ |
| formalizes | reconstruction of integers from residue classes ⓘ |
| hasAlternativeName | CRT ⓘ |
| hasFormulation |
integer arithmetic formulation
ⓘ
ring-theoretic formulation ⓘ |
| hasGeneralization |
Chinese remainder theorem for ideals
ⓘ
linked to:
Chinese remainder theorem
Chinese remainder theorem for rings ⓘ
linked to:
Chinese remainder theorem
Chinese remainder theorem in commutative algebra ⓘ |
| hasHistoricalOrigin | ancient Chinese mathematics ⓘ |
| hasProofTechnique |
constructive proof using modular inverses
ⓘ
proof using Bezout coefficients ⓘ proof using ring isomorphisms ⓘ |
| implies | isomorphism Z/(n1⋯nk)Z ≅ Z/n1Z × ⋯ × Z/nkZ for pairwise coprime ni ⓘ |
| involves |
congruence relations
ⓘ
moduli ⓘ pairwise coprime integers ⓘ |
| provides |
existence conditions for simultaneous solutions
ⓘ
uniqueness conditions for simultaneous solutions modulo the product of the moduli ⓘ |
| relatedTo |
Bezout's identity
ⓘ
linked to:
Euclidean algorithm for polynomials
Euclidean algorithm ⓘ
linked to:
Euclidean algorithm for polynomials
direct product of rings ⓘ modular inverses ⓘ ring isomorphisms ⓘ |
| requiresCondition | pairwise coprimality of moduli ⓘ |
| states |
if moduli are pairwise coprime then a simultaneous solution exists
ⓘ
if moduli are pairwise coprime then the solution is unique modulo the product of the moduli ⓘ |
| subfield | elementary number theory ⓘ |
| usedFor |
calendar calculations
ⓘ
clock arithmetic problems ⓘ solving congruence-based puzzles ⓘ speeding up modular exponentiation ⓘ |
| usedIn |
RSA algorithm
ⓘ
linked to:
RSA
algorithmic Chinese remainder representation of integers ⓘ coding theory ⓘ computational number theory ⓘ computer algebra systems ⓘ cryptography ⓘ integer reconstruction from residues ⓘ parallel computation of modular operations ⓘ |
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: Chinese remainder theorem Description of subject: The Chinese remainder theorem is a fundamental result in number theory that provides conditions and a method for solving systems of simultaneous congruences with pairwise coprime moduli.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Chinese remainder theorem
linked to: Chinese remainder theorem
linked to: Chinese remainder theorem
linked to: Chinese remainder theorem