Diophantine equations
E629500
Diophantine equations are polynomial equations for which only integer or rational solutions are sought, forming a central and often notoriously difficult area of number theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Diophantine equations canonical | 4 |
| Diophantine Equations | 1 |
| Thue-type equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6938598 — 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: Diophantine equations Context triple: [Unsolved Problems in Number Theory, coversTopic, Diophantine equations]
-
A.
Diophantine geometry
Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
-
B.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
-
C.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
-
D.
Erdős–Moser equation
The Erdős–Moser equation is a famous unsolved Diophantine equation in number theory that asks whether 1^k + 2^k + ... + (m−1)^k = m^k has any integer solutions beyond the trivial case (k, m) = (1, 2).
-
E.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
- 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: Diophantine equations Target entity description: Diophantine equations are polynomial equations for which only integer or rational solutions are sought, forming a central and often notoriously difficult area of number theory.
-
A.
Diophantine geometry
Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
-
B.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
-
C.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
-
D.
Erdős–Moser equation
The Erdős–Moser equation is a famous unsolved Diophantine equation in number theory that asks whether 1^k + 2^k + ... + (m−1)^k = m^k has any integer solutions beyond the trivial case (k, m) = (1, 2).
-
E.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
- F. None of above. chosen
Statements (54)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
topic in number theory ⓘ type of equation ⓘ |
| centralIn | number theory ⓘ |
| decidabilityResult | no algorithm exists to decide solvability over integers (Hilbert's tenth problem) ⓘ |
| distinguishedFrom |
complex-valued polynomial equations
ⓘ
real-valued polynomial equations ⓘ |
| field | number theory ⓘ |
| hasExample |
Fermat equation x^n + y^n = z^n
NERFINISHED
ⓘ
Markov equation x^2 + y^2 + z^2 = 3xyz NERFINISHED ⓘ Mordell equation y^2 = x^3 + k ⓘ Pell equation NERFINISHED ⓘ Pythagorean triple equation x^2 + y^2 = z^2 ⓘ Thue equation NERFINISHED ⓘ unit equation x + y = 1 in S-units ⓘ |
| hasSubcategory |
Diophantine approximation
ⓘ
Diophantine inequalities ⓘ Diophantine sets NERFINISHED ⓘ exponential Diophantine equations ⓘ higher-degree Diophantine equations ⓘ linear Diophantine equations ⓘ quadratic Diophantine equations ⓘ |
| involves | polynomial equations ⓘ |
| knownFor | difficulty ⓘ |
| namedAfter | Diophantus of Alexandria NERFINISHED ⓘ |
| relatedTo |
Birch and Swinnerton-Dyer conjecture
NERFINISHED
ⓘ
Diophantine sets NERFINISHED ⓘ Faltings' theorem NERFINISHED ⓘ Fermat's Last Theorem NERFINISHED ⓘ Galois representations NERFINISHED ⓘ Hasse principle ⓘ Hilbert's tenth problem NERFINISHED ⓘ Matiyasevich's theorem NERFINISHED ⓘ Mordell conjecture NERFINISHED ⓘ Siegel's theorem NERFINISHED ⓘ abc conjecture NERFINISHED ⓘ elliptic curves ⓘ height functions ⓘ local-global principle ⓘ modular forms ⓘ rational points on varieties ⓘ |
| solutionDomain |
integers
ⓘ
rational numbers ⓘ |
| solvabilityProperty | undecidable in general ⓘ |
| studiedIn |
ancient Greek mathematics
ⓘ
modern mathematics ⓘ |
| subfieldOf |
algebraic number theory
ⓘ
arithmetic geometry ⓘ |
| usesMethod |
algebraic geometry
ⓘ
computational number theory ⓘ congruences ⓘ geometry of numbers ⓘ p-adic methods ⓘ transcendence 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: Diophantine equations Description of subject: Diophantine equations are polynomial equations for which only integer or rational solutions are sought, forming a central and often notoriously difficult area of number theory.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Diophantine Equations
this entity surface form:
Thue-type equation