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Fermat–Catalan conjecture
Generalization of Fermat's Last Theorem and of Catalan's conjecture, From Wikipedia, the free encyclopedia
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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
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
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), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g. .
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Known solutions
Summarize
Perspective
As of 2015 the following ten solutions to equation (1) which meet the criteria of equation (2) are known:[1][2]
- (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).
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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 (2), only finitely many coprime triples (a, b, c) solving (1) exist.[3][4]: 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.[5]
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.
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See also
- Sums of powers, a list of related conjectures and theorems
References
External links
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