Erdős–Straus conjecture
E554304
The Erdős–Straus conjecture is an unsolved problem in number theory asserting that for every integer n ≥ 2, the fraction 4/n can be expressed as a sum of three unit fractions.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Erdős–Straus conjecture canonical | 2 |
| Egyption fraction conjectures | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5896711 — 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: Erdős–Straus conjecture Context triple: [Pál Erdős, knownFor, Erdős–Straus conjecture]
-
A.
Beal conjecture
The Beal conjecture is an unsolved problem in number theory proposing that if A^x + B^y = C^z with A, B, C, x, y, z positive integers and exponents greater than 2, then A, B, and C must share a common prime factor.
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B.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
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C.
Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
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D.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
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E.
Goldbach conjecture
The Goldbach conjecture is a famous unsolved problem in number theory asserting that every even integer greater than 2 can be expressed as the sum of two prime 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: Erdős–Straus conjecture Target entity description: The Erdős–Straus conjecture is an unsolved problem in number theory asserting that for every integer n ≥ 2, the fraction 4/n can be expressed as a sum of three unit fractions.
-
A.
Beal conjecture
The Beal conjecture is an unsolved problem in number theory proposing that if A^x + B^y = C^z with A, B, C, x, y, z positive integers and exponents greater than 2, then A, B, and C must share a common prime factor.
-
B.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
C.
Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
-
D.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
E.
Goldbach conjecture
The Goldbach conjecture is a famous unsolved problem in number theory asserting that every even integer greater than 2 can be expressed as the sum of two prime numbers.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf | mathematical conjecture ⓘ |
| concerns |
Egyptian fractions
ⓘ
linked to:
Egyptian mathematics
representation of rational numbers as sums of unit fractions ⓘ |
| difficulty | considered hard in elementary number theory ⓘ |
| domainOfVariable | positive integers n ≥ 2 ⓘ |
| field | number theory ⓘ |
| formalStatement | For every integer n ≥ 2, there exist positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z. ⓘ |
| hasAlternativeFormulation | For every n ≥ 2, the Diophantine equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers x, y, z. ⓘ |
| hasComputationalVerification | verified for all n up to very large bounds by computer search ⓘ |
| hasForm | 4/n = 1/x + 1/y + 1/z with x, y, z ∈ ℕ ⓘ |
| hasNotation | 4/n = 1/a + 1/b + 1/c ⓘ |
| hasPartialResults |
proved for all n below large explicit bounds
ⓘ
proved for all n in many congruence classes modulo various integers ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| impliedBy | truth for all prime n ≥ 2 ⓘ |
| influenced | research on Egyptian fractions ⓘ |
| involvesEquation | 4nxyz = nxy + nxz + nyz ⓘ |
| isSpecialCaseOf | problems on expressing rational numbers as sums of few unit fractions ⓘ |
| languageOfOriginalPublication | English ⓘ |
| literatureType | research papers in analytic and elementary number theory ⓘ |
| motivation | understanding structure of Egyptian fraction representations ⓘ |
| namedAfter |
Ernst G. Straus
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| namedBy |
Paul Erdős and Ernst G. Straus
ⓘ
linked to:
Pál Erdős
|
| openProblemIn |
Diophantine equations
ⓘ
elementary number theory ⓘ |
| reduction | It suffices to prove the conjecture for prime n. ⓘ |
| relatedConcept |
Egyptian fraction decomposition
ⓘ
linked to:
Egyptian mathematics
unit fraction ⓘ |
| relatedConjecture |
Egyption fraction conjectures
ⓘ
linked to:
Erdős–Straus conjecture
|
| statement | For every integer n ≥ 2, the fraction 4/n can be expressed as a sum of three unit fractions. ⓘ |
| status | open ⓘ |
| subfield |
Diophantine analysis
ⓘ
linked to:
Diophantine equations
additive number theory ⓘ |
| topic | representation of 4/n as sum of three unit fractions ⓘ |
| typicalMethod |
case analysis by congruence classes of n
ⓘ
computer-assisted search for representations ⓘ |
| unknownFor | all n in general ⓘ |
| usesConcept |
Diophantine equations
ⓘ
computational number theory ⓘ modular arithmetic ⓘ |
| variableConstraint |
n is an integer with n ≥ 2
ⓘ
x, y, z are positive integers ⓘ |
| yearProposed | 1948 ⓘ |
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: Erdős–Straus conjecture Description of subject: The Erdős–Straus conjecture is an unsolved problem in number theory asserting that for every integer n ≥ 2, the fraction 4/n can be expressed as a sum of three unit fractions.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Erdős–Straus conjecture