In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation
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(1) |
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
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(2) |
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) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples).
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 (2), only finitely many coprime triples (a, b, c) solving (1) exist.[2][3]: p. 64 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.[4]
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.
Pomerance, Carl (2008), "Computational Number Theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362, ISBN 978-0-691-11880-2.