Fermat–Catalan conjecture

In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation
has only finitely many solutions with distinct triplets of values where a, b, c are positive coprime integers and m, n, k are positive integers satisfying
The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1, with m=''n=k''=2, and e.g..

Known solutions

As of 2024, the following ten solutions to equation which meet the criteria of equation are known:
The first of these is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of, these solutions only give a single triplet of values.

Partial results

It is known by the Darmon–Granville theorem, which uses Faltings's theorem, that for any fixed choice of positive integers m, n and k satisfying, only finitely many coprime triples solving exist. However, the full Fermat–Catalan conjecture is stronger as it allows for the exponents m, n and k to vary.
The abc conjecture implies the Fermat–Catalan conjecture.
For a list of results for impossible combinations of exponents, see Beal conjecture#Partial results. Beal's conjecture is true if and only if all Fermat–Catalan solutions have m = 2, n = 2, or k = 2.
Poonen et al. list exponent triples where the solutions have been determined:
,
,
,
for prime 7<q<1000 with q≠31,
,
,
,
for prime q≥211,
,
,
,
,
for 17≤q≤10000,
,
,
.
For each of these exponent triples, if there is some solution at all, it is listed among those in section.
Sikora partially used the cluster computers at the Center for Computational Research at University at Buffalo to test all tuples such that min ≤ 113 and a^m, bⁿ, c^k < M_min, where M₂ = 2⁷¹, M₃ = 2⁸⁰, M₄ = 2¹⁰⁰, and M₅ =... = M₁₁₃ = 2¹¹³. He did not find any other solution than those above.