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 (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck) 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 = am + bn), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g. .

Known solutions As of 2024, the following ten solutions to equation (1) which meet the criteria of equation (2) are known:. Here: p.61: "the larger [solutions] were found by a computer search performed on Fermat day at Utrecht in November 1993 ... Notice that in each solution an exponent 2 occurs."

(for to satisfy Eq. 2)

The first of these (1m + 23 = 32) 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 (1) (since one can pick any m for m > 6), these solutions only give a single triplet of values (am, bn, ck).

Partial results It is known by the Darmon–Granville 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 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. Here: p.3, table 1. list exponent triples where the solutions have been determined:The notation "{p,q,r}" means that the solutions have been determined for every permutation of (p,q,r). {2,3,7}, {2,3,8}, {2,3,9}, {2,2q,3} for prime 7<q<1000 with q≠31, {2,4,5}, {2,4,6}, (2,4,7), (2,4,q) for prime q≥211, (2,n,4),"follows easily from" the special case (4,n,4) was done earlier in {2,n,n}, {3,3,4}, {3,3,5}, {3,3,q} for 17≤q≤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 am, bn, ck < Mmin(m,n,k), where M2 = 271, M3 = 280, M4 = 2100, and M5 = ... = M113 = 2113. He did not find any other solution than those above.For example, the five known large solutions were all reproduced during the test for min(m,n,k) = 2, where am, bn, and ck were considered up to 271.

Fermat–Catalan conjecture

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