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.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | mathematical conjecture ⓘ |
| alsoKnownAs |
Beal’s conjecture
NERFINISHED
ⓘ
Mauldin’s conjecture NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| hasCounterexampleStatus | no counterexample known ⓘ |
| hasPrize | Beal Prize NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| prizeRequirement |
solution must appear in a refereed mathematics journal
ⓘ
solution must be accepted by the mathematical community ⓘ |
| prizeSponsor | American Mathematical Society NERFINISHED ⓘ |
| proposedBy | Andrew Beal NERFINISHED ⓘ |
| proposerNationality | American ⓘ |
| proposerOccupation | banker ⓘ |
| publicationVenueForStatement | Notices of the American Mathematical Society NERFINISHED ⓘ |
| relatedTo |
Catalan’s conjecture
NERFINISHED
ⓘ
Fermat’s Last Theorem NERFINISHED ⓘ abc conjecture NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.