Beal's conjecture is a claim in number theory stating that if A^x + B^y = C^z, where A, B, C, x, y and z are positive integers and x, y and z are each greater than 2, then A, B and C must share a common prime factor. It was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while he was investigating generalizations of Fermat's Last Theorem. The conjecture remains unproven and open.
Facts
StatementIf A, B, C, x, y and z are positive integers with x, y and z each greater than 2, and A to the x plus B to the y equals C to the z, then A, B and C share a common prime factor. 1 Proposed Year Prize StatusAndrew Beal offered a cash prize for a proof or disproof of the conjecture, beginning at 5000 US dollars in 1997 and raised over the following decade to 50000 US dollars, and since raised further to 1,000,000 US dollars; the American Mathematical Society holds the million dollar prize in trust until the conjecture is solved. 1 Progress Toward ResolutionFermat's Last Theorem, the case where x, y and z are all equal and at least 3, was proven by Andrew Wiles in 1994; several specific exponent triples such as (2,3,7), (2,3,8), (2,3,9) and (2,3,10) have also been proven through work by Poonen, Bennett, Bruin, Siksek and others. Peter Norvig ran a computational search excluding all solutions with each of x, y, z at most 7 and each of A, B, C at most 250,000, as well as with each of x, y, z at most 100 and each of A, B, C at most 10,000. The general conjecture remains open. 1 Classification
Resolution Status Prize Status
Prize Status (category)Personally-Funded Prize 1 Connections
In Branch
Source Beal Conjecture (Wikipedia)
Posed By
Sources
1. Beal Conjecture (Wikipedia)
Wikimedia FoundationStatement 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.
History section
The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.
Prize section
The American Mathematical Society (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved.
Partial results section, computational search
he excluded all possible solutions having each of x, y, z ≤ 7 and each of A, B, C ≤ 250,000, as well as possible solutions having each of x, y, z ≤ 100 and each of A, B, C ≤ 10,000.
- Lead section
- In Branch: Number Theory, Lead sentence
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