Beal conjecture

History

This conjecture was made by Andrew Beal, a mathematics hobbyist, while investigating Fermat's last theorem.

Statement

Consider the equation:

$\!a^{m}+b^{n}=c^{k}$.

The Beal conjecture (also called Beal's conjecture) states the following equivalent things:

1. This equation has no solutions for $a,b,c$ pairwise relatively prime positive integers, and $m,n,k$ all natural numbers greater than $2$.
2. This equation has no solutions for $a,b,c$ pairwise relatively prime integers (all nonzero) and $m,n,k$ all natural numbers greater than $2$.

=Stronger facts and conjectures

Weaker facts and conjectures

• Fermat's last theorem: This is the special case $m=n=k$. This was conjectured by Fermat and proved by Wiles, building on work by several mathematicians in between.

Other related facts

• Euler's false attempted generalization of Fermat's last theorem
• Fermat-Catalan conjecture: With somewhat weaker hypotheses on the exponents, it claims that there are only finitely many solutions.

Failure of slight modifications of the conjecture

• Beal conjecture fails over Gaussian integers
• Beal conjecture fails if condition of relative primality is dropped: In fact, there are easy-to-parametrize families of solutions.
• Analogue of Beal conjecture with relative primality condition on exponents instead of bases fails

External links

• Beal Conjecture page
• Information about Beal Conjecture and prize
• Peter Norvig's explanation of a computer program for the Beal conjecture