Beal's conjecture
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: A Texas millionaire banker named Andre Beal has offered a $75,000 cash prize to the first person to prove (or disprove) his conjecture:
Beal's Conjecture:
If xm + yn = zr where x, y, z m, n and r are all positive integers, and m, n and r are greater than two, then x, y, and z have a common factor (greater than one).
Clearly Fermat's Last Theorem is a special case of this conjecture, so if we could find some easy way to transform this into Fermat's Last Theorem, then we would be done via Wiles proof.

If we do not require the exponents to be greater than two, then there are infinitely many solutions such as 11+23=32, 25+72=34, and all Pythagorean triples. Also there are infinitely many soultions for which x, y and z are not relatively prime such as 2n+2n = 2n+1.

It is known that for any set of three exponents m, n, and r, each greater than two, there can be at most finitely many solutions. But is this finite number zero? See the link below to find out where to send your proof or disproof!

See Also: FermatsLastTheorem, CatalansProblem

Related pages (outside of this work)




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