In number theory, the FermatâÂÂCatalan 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 (a, b, c, m, n, k) with distinct triplets of values (a<sup>m</sup>, b<sup>n</sup>, c<sup>k</sup>) 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 (for any a, b, m, and n and with c = a<sup>m</sup> + b<sup>n</sup>), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g. .
As of 2024, the following ten solutions to equation (1) which meet the criteria of equation (2) are known:
The first of these (1<sup>m</sup> + 2<sup>3</sup> = 3<sup>2</sup>) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda MihÃÂilescu. While this case leads to infinitely many solutions of (1) (since one can pick any m for m > 6), these solutions only give a single triplet of values (a<sup>m</sup>, b<sup>n</sup>, c<sup>k</sup>).
It is known by the DarmonâÂÂGranville theorem, which uses Faltings' theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (a, b, c) solving (1) exist. However, the full FermatâÂÂCatalan conjecture is stronger as it allows for the exponents m, n and k to vary.
The abc conjecture implies the FermatâÂÂCatalan 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 FermatâÂÂCatalan solutions have m = 2, n = 2, or k = 2.
Poonen et al. list exponent triples where the solutions have been determined: {2,3,7}, {2,3,8}, {2,3,9}, {2,2q,3} for prime 7<q<1000 with qâ 31, {2,4,5}, {2,4,6}, (2,4,7), (2,4,q) for prime qâÂÂ¥211, (2,n,4), {2,n,n}, {3,3,4}, {3,3,5}, {3,3,q} for 17â¤qâ¤10000, {3,n,n}, {2n,2n,5}, {n,n,n}. 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 (a,b,c,m,n,k) such that min(m,n,k) ⤠113 and a<sup>m</sup>, b<sup>n</sup>, c<sup>k</sup> < M<sub>min(m,n,k)</sub>, where M<sub>2</sub> = 2<sup>71</sup>, M<sub>3</sub> = 2<sup>80</sup>, M<sub>4</sub> = 2<sup>100</sup>, and M<sub>5</sub> = ... = M<sub>113</sub> = 2<sup>113</sup>. He did not find any other solution than those above.