Fermat-Catalan conjecture

Statement

The equation:

with positive integers such that are relatively prime and:

has only finitely many solutions, where, in the special case that or , we count all solutions with distinct values of the corresponding exponent as equivalent.

(Note that sometimes the conjecture is written with strict inequality . This is an equivalent formulation because in all the cases of exact equality with , it is known that there are no solutions).

Status

The conjecture is open. A certain finite list of solutions has been found, but the conjecture does not claim that this is the complete list. Note that to avoid unnecessary duplication, we do not list the mirror solutions to each solution thatcan be obtained by interchanging with and with . The solutions found so far are:

Equation										Value of	Comment
,	1	2	3	-	3	2	1	8	9		Catalan's conjecture states that this is the only solution with . That conjecture has been proved.
	2	7	3	5	2	4	32	49	81	0.95 = 19/20
	13	7	2	2	3	9	169	343	512	17/18
	2	17	71	7	3	2	128	4913	5041	41/42
	3	11	122	5	4	2	243	14641	14884	0.95 = 19/20
	33	1549034	15613	8	2	3	1406408618241	2399506333156	3805914951397	23/24
	1414	2213459	65	3	2	7	2827145944	4899400744681	4902227890625	41/42
	9262	15312283	113	3	2	7	794537372728	234466010672089	235260548044817	41/42
	17	76271	21063928	7	3	2	410338673	443688652450511	443689062789184	41/42
	43	96222	30042907	8	3	2	11688200277601	890888060733048	902576261010649	23/24

Relation with other facts/conjectures

Stronger facts and conjectures

Conjecture	Statement	Status	Proof of implication
abc conjecture	For every ,there exists such that for all ,	open	abc conjecture implies Fermat-Catalan conjecture

Other related conjectures and facts

Conjecture/fact	Statement	Status	Nature of relationship
Catalan's conjecture	The only solution to for and positive is	proved	If a complete set of solutions to the Fermat-Catalan conjecture is found and proved to be complete, Catalan's conjecture would be a special case.
Fermat's last theorem	has no solutions for and positive integers	proved	The Fermat-Catalan conjecture would imply that there are only finitely many solutions. The Fermat-Catalan conjecture along with an explicit list of solutions would prove Fermat's last theorem.
Beal conjecture	has no solutions for relatively prime with all at least	The Fermat-Catalan conjecture would imply that there are only finitely many solutions. If none of the solutions to the Fermat-Catalan problem provides a counterexample to the Beal conjecture, this would prove the Beal conjecture.
Faltings' theorem	This states that certain kinds of algebraic curves have only finitely many rational points	proved	Faltings' theorem implies that for fixed choice of , there are only finitely many solutions. The Fermat-Catalan conjecture goes further: it says that for all but finitely many permissible choices of , there are no solutions.