Beal conjecture
E530308
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Beal conjecture canonical | 1 |
| Beal’s conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570543 — 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: Beal conjecture Context triple: [Fermat's Last Theorem, relatedProblem, Beal conjecture]
-
A.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
-
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.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
D.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
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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: Beal conjecture Target entity description: 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.
-
A.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
-
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.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
D.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
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 (46)
| Predicate | Object |
|---|---|
| instanceOf | mathematical conjecture ⓘ |
| alsoKnownAs |
Beal’s conjecture
ⓘ
linked to:
Beal conjecture
Mauldin’s conjecture ⓘ |
| computationalVerification | verified for many ranges of parameters by computer search ⓘ |
| conclusion | A,B,C share a common prime factor ⓘ |
| conditionOnExponents |
x > 2
ⓘ
y > 2 ⓘ z > 2 ⓘ |
| conditionOnVariables | A,B,C,x,y,z are positive integers ⓘ |
| difficulty | considered very hard ⓘ |
| equationForm | A^x + B^y = C^z ⓘ |
| field | number theory ⓘ |
| firstPrizeAnnouncementYear | 1997 ⓘ |
| formulation | If A^x + B^y = C^z with A,B,C,x,y,z positive integers and x,y,z > 2, then A,B,C have a common prime factor ⓘ |
| generalizes |
Fermat’s Last Theorem
ⓘ
linked to:
Fermat's Last Theorem
|
| hasCounterexampleStatus | no counterexample known ⓘ |
| hasPrize | Beal Prize ⓘ |
| implies | no nontrivial coprime integer solutions to A^x + B^y = C^z with x,y,z > 2 ⓘ |
| involvesConcept |
coprime integers
ⓘ
exponential Diophantine equation ⓘ greatest common divisor ⓘ prime factorization ⓘ prime numbers ⓘ |
| logicalForm | implication from existence of solution to existence of common prime factor ⓘ |
| openProblemList | listed among major unsolved problems in number theory ⓘ |
| prizeAmount | 1000000 US dollars ⓘ |
| prizeCondition | awarded for a published proof or counterexample ⓘ |
| prizeDonor | Andrew Beal ⓘ |
| prizeRequirement |
solution must appear in a refereed mathematics journal
ⓘ
solution must be accepted by the mathematical community ⓘ |
| prizeSponsor | American Mathematical Society ⓘ |
| proposedBy | Andrew Beal ⓘ |
| proposerNationality | American ⓘ |
| proposerOccupation | banker ⓘ |
| publicationVenueForStatement | Notices of the American Mathematical Society ⓘ |
| relatedTo |
Catalan’s conjecture
ⓘ
Fermat’s Last Theorem ⓘ
linked to:
Fermat's Last Theorem
abc conjecture ⓘ |
| researchArea | exponential Diophantine equations ⓘ |
| researchedBy | number theorists ⓘ |
| restrictionOnABC | A,B,C are pairwise coprime only if no solution with x,y,z > 2 exists ⓘ |
| specialCase | When x = y = z, equation resembles generalized Fermat-type equations ⓘ |
| status |
open problem
ⓘ
unproven ⓘ |
| subfield | Diophantine equations ⓘ |
| yearProposed | 1993 ⓘ |
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: Beal conjecture Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Beal conjecture