# Beal's conjecture

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
x+^{m}y=^{n}zwhere^{r}x,y,zm,nandrare all positive integers, andm,nandrare greater than two, thenx,y, andzhave 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
1^{1}+2^{3}=3^{2}, 2^{5}+7^{2}=3^{4}, and all
Pythagorean triples. Also there are infinitely many solutions for which *x*, *y* and *z*
are not relatively prime such as 2^{n}+2^{n} = 2^{n+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)