User:Noosfractal/Beal's Conjecture

While investigating mathematical forms associated with Fermat's last theorem, Andrew Beal noticed a remarkable pattern, leading him to conjecture:

Beal's conjecture: if x^m + yⁿ = z^r, where m, n, r, x, y and z are positive integers with m, n, r > 2, then x, y and z must have a common prime factor.

For example, the solution 3³ + 6³ = 3⁵ has bases with a common factor of 3, and the solution 7⁶ + 7⁷ = 98³ has bases with a common factor of 7. As of 2005, there are no known counterexamples.

A reward of $100,000 is offered for a proof or disproof of the conjecture.

References

• http://www.bealconjecture.com/
• A search for counterexamples - http://www.norvig.com/beal.html
• http://planetmath.org/encyclopedia/BealsConjecture.html