# Beal conjecture

From Number

## Contents

## History

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

## Statement

Consider the equation:

.

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

- This equation has no solutions for pairwise relatively prime positive integers, and all natural numbers greater than .
- This equation has no solutions for pairwise relatively prime integers (all nonzero) and all natural numbers greater than .

## Related facts

### Weaker facts and conjectures

- Fermat's last theorem: This is the special case . This was conjectured by Fermat and proved by Wiles, building on work by several mathematicians in between.

- 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