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Conjecture

Beal Conjecture

BEEL kon-JEK-cher (rhymes with wheel; named for Andrew Beal)
Also Known As Tijdeman-Zagier Conjecture
Number Theory

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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
If 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
1993 1
Prize Status
Not 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 Resolution
Partial 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
Prize Status
Prize Status (category)
Millennium Prize Problem 1
Prize Status (category)
Personally-Funded Prize 1
Classification
Resolution Status
Open 1
Connections

Associated With

Fermat's Last Theorem, Theorems

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.

Source Beal Conjecture (Wikipedia)

In Branch

Number Theory, Branches of Mathematics
Source Beal Conjecture (Wikipedia)

Open Questions

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?, Open Questions
Source Beal Conjecture (Wikipedia)

Posed By

Andrew Beal, Mathematicians

Beal formulated the conjecture in 1993 and has personally funded its prize since 1997.

Source Beal Conjecture (Wikipedia)
Sources
1. Beal Conjecture (Wikipedia)
Wikimedia Foundation
  • 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.
  • Introductory paragraph
    The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.
  • 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.
  • Categories
    Unsolved problem in mathematics.
  • Lead section
  • In Branch: Number Theory, Introductory sentence
    The Beal conjecture is the following conjecture in number theory
  • Posed By: Andrew Beal, Introduction section
    The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations
  • 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.
View the Source
Wolfram MathWorld
Wolfram Research, Inc.https://mathworld.wolfram.com/BealsConjecture.html
Quote, https://mathworld.wolfram.com/BealsConjecture.html
a cash prize of $1000000 has been offered for its proof or a counterexample
View the Source
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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