Beal's conjecture

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In 1993, Andrew Beal made the following conjecture:

Conjecture (Beal's conjecture, 1993). (Andrew Beal)

If

$A^{x}+B^{y}=C^{z},\,$

where $\scriptstyle A,\,B,\,C,\,x,\,y\,$ and $\scriptstyle z\,$ are positive integers and $\scriptstyle x,y\,$ and $\scriptstyle z\,$ are all greater than 2, then $\scriptstyle A,B\,$ and $\scriptstyle C\,$ must have a common prime factor (i.e. $\scriptstyle \gcd(A,\,B,\,C)\,>\,1\,$).

Or, slightly restated, the equation

$A^{x}+B^{y}=C^{z},\,$

has no solution in positive integers $\scriptstyle A,\,B,\,C,\,x,\,y\,$ and $\scriptstyle z\,$ with $\scriptstyle x,y\,$ and $\scriptstyle z\,$ all greater than 2 and $\scriptstyle A,B\,$ and $\scriptstyle C\,$ being 3-wise coprime (i.e. $\scriptstyle \gcd(A,\,B,\,C)\,=\,1\,$).

The Beal conjecture is sometimes referred to as "Beal's conjecture", "Beal's problem" or the "Beal problem."

It turns out that very similar conjectures have been made over the years. In fact, Brun in his 1914 paper states several similar problems [1].

A search for counterexamples

There is a $100,000 prize for the first proof or disproof of the conjecture. The conjecture is obviously related to Fermat's last theorem, which was proved true by Andrew Wiles in 1994. The majority of mathematicians competent to judge seem to believe that it likely is true.

See also

• Fermat's last theorem
• Abc conjecture

References

1. Brun, V. (1914). “Über Hypothesenbildung”. Arc. Math. Naturvidenskab 34: pp. 1–14. 

External links

• The Beal Conjecture.
• R. Daniel Mauldin, The Beal Conjecture and Prize.