Conjectures
Beal Conjecture
BEEL kon-JEK-cher (rhymes with wheel; named for Andrew Beal)
Also Known As Tijdeman-Zagier Conjecture
Number Theory
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Formulated in 1993 by Andrew Beal, a Texas banker and amateur mathematician, while he was investigating generalizations of Fermat's Last Theorem. Unlike most named conjectures in this atlas, it was proposed outside professional mathematics and is sustained by private money rather than an institution: Beal himself funds the prize for its proof or refutation.
Facts
StatementIf A to the x plus B to the y equals C to the z, where A, B, C, x, y and z are positive integers with x, y and z all greater than two, then A, B and C share a common prime factor. 1 Proposed Year Prize StatusNot a Millennium Prize Problem. Andrew Beal has personally funded a prize for a published proof or counterexample, raised over time from five thousand dollars in 1997 to one million dollars, held in trust by the American Mathematical Society until the conjecture is solved. 1 Progress Toward ResolutionPartial results establish the conjecture for many specific combinations of exponents, but no complete proof or counterexample is known and it remains an unsolved problem in mathematics. 1 Cross-Tradition Connections
Associated With
Fermat's Last Theorem is the special case of the Beal conjecture where the three exponents are forced equal, per Wikipedia's Relation to other conjectures section.
In Branch
Posed By
Beal formulated the conjecture in 1993 and has personally funded its prize since 1997.
Sources
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationStatement sectionQuote, Statement section
If A^x + B^y = C^z, where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor.
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationIntroductory paragraphQuote, Introductory paragraph
The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationPrize sectionQuote, Prize section
For a published proof or counterexample, banker Andrew Beal initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years, but has since raised it to US $1,000,000.
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationCategoriesQuote, Categories
Unsolved problem in mathematics.
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationIn Branch: Number Theory, Introductory sentenceQuote, In Branch: Number Theory, Introductory sentence
The Beal conjecture is the following conjecture in number theory
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationPosed By: Andrew Beal, Introduction sectionQuote, Posed By: Andrew Beal, Introduction section
The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationAssociated With: Fermat's Last Theorem, Relation to other conjectures sectionQuote, Associated With: Fermat's Last Theorem, Relation to other conjectures section
Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.
Wolfram MathWorld
Wolfram Research, Inc.https://mathworld.wolfram.com/BealsConjecture.htmlQuote, https://mathworld.wolfram.com/BealsConjecture.html
a cash prize of $1000000 has been offered for its proof or a counterexample
Open Questions (1 open question)
Whenever A to the x plus B to the y equals C to the z, with x, y and z all greater than two, do A, B and C really always share a common prime factor?
No complete proof or counterexample has been found since Andrew Beal proposed the conjecture in 1993. Partial results confirm it for many specific combinations of exponents, but a general argument covering every combination, or a single counterexample disproving it, has eluded both professional and amateur attempts despite the million dollar prize Beal has offered.
What would resolve this A general proof covering every valid combination of A, B, C, x, y and z, or a single confirmed counterexample: one solution where A, B and C share no common prime factor.
Number theoryBeal Conjecture (Wikipedia)
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