# Fermat–Catalan conjecture

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In number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name. The conjecture states that the equation

${\displaystyle a^{m}+b^{n}=c^{k}\quad }$

(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

${\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}<1.}$

(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).

## Known solutions

As of 2015, the following ten solutions to (1) are known:[1]

${\displaystyle 1^{m}+2^{3}=3^{2}\;}$
${\displaystyle 1414^{3}+2213459^{2}=65^{7}\;}$
${\displaystyle 9262^{3}+15312283^{2}=113^{7}\;}$
${\displaystyle 7^{3}+13^{2}=2^{9}\;}$
${\displaystyle 2^{5}+7^{2}=3^{4}\;}$
${\displaystyle 3^{5}+11^{4}=122^{2}\;}$
${\displaystyle 2^{7}+17^{3}=71^{2}\;}$
${\displaystyle 17^{7}+76271^{3}=21063928^{2}\;}$
${\displaystyle 33^{8}+1549034^{2}=15613^{3}\;}$
${\displaystyle 43^{8}+96222^{3}=30042907^{2}\;}$

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's theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (abc) 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.

## References

1. ^ Pomerance, Carl (2008), "Computational Number Theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre, The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362, ISBN 978-0-691-11880-2.
2. ^ Darmon, H.; Granville, A. (1995). "On the equations zm = F(x, y) and Axp + Byq = Czr". Bulletin of the London Mathematical Society. 27: 513–43. doi:10.1112/blms/27.6.513.
3. ^ Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review. 1 (1).
4. ^ Waldschmidt, Michel (2015). "Lecture on the ${\displaystyle abc}$ conjecture and some of its consequences". Mathematics in the 21st century (PDF). Springer Proc. Math. Stat. 98. Basel: Springer. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. MR 3298238.