The conjecture that there are only finitely many triples of relatively prime integer powers 
, 
, 
for which
 |
(1)
|
with
 |
(2)
|
Darmon and Merel (1997) have shown that there are no relatively prime solutions 
with 
.
Ten solutions are known,
 |
(3)
|
for 
,
and
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
(Mauldin 1997).
The following table summarizes known solutions (Poonen et al. 2005). Any remaining solutions would satisfy the Tijdeman-Zagier conjecture, also known popularly as Beal's conjecture (Elkies 2007).
exponents  | reference |
| (2, 3, 7) | Poonen et al. (2005) |
 | Wiles |
| (2, 3, 8), (2, 3, 9), (2, 4, 5), | Bruin (2004) |
| (2, 4, 6), (3, 3, 4), (3, 3, 5) | |
| (2, 4, 7) | Ghioca |
 ,  | Darmon-Merel |
 | Bennett |
 | Bennett-Skinner |
It is not known if the analogous conjecture for 
, 
, and 
Gaussian integers holds.
Known solutions include
 |  |  |
(13)
|
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
(E. Pegg Jr., pers. comm., Mar. 30, 2002 and Mar. 26, 2025).
and 
." Bull. London Math. Soc. 27,
513-543, 1995.Darmon, H. and Merel, L. "Winding Quotients and Some
Variants of Fermat's Last Theorem." J. reine angew. Math. 490,
81-100, 1997.Elkies, N. "The ABCs of Number Theory." Harvard
Math. Rev. 1, 64-76, 2007.Mauldin, R. D. "A Generalization
of Fermat's Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer.
Math. Soc. 44, 1436-1437, 1997.Poonen, B.; Schaefer, E. F.;
and Stoll, M. "Twists of 
and Primitive Solutions to 
." 10 Aug 2005.