Fermat–Catalan conjecture

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

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

(1)

has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (a^m, bⁿ, c^k) 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 = a^m + 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 equation (1) which meet the criteria of equation (2) are known:^[1]

${\displaystyle 1^{m}+2^{3}=3^{2}\;}$  (for ${\displaystyle m>6}$  to satisfy Eq. 2)

${\displaystyle 2^{5}+7^{2}=3^{4}\;}$ 

${\displaystyle 7^{3}+13^{2}=2^{9}\;}$ 

${\displaystyle 2^{7}+17^{3}=71^{2}\;}$ 

${\displaystyle 3^{5}+11^{4}=122^{2}\;}$ 

${\displaystyle 33^{8}+1549034^{2}=15613^{3}\;}$ 

${\displaystyle 1414^{3}+2213459^{2}=65^{7}\;}$ 

${\displaystyle 9262^{3}[phone]^{2}=113^{7}\;}$ 

${\displaystyle 17^{7}+76271^{3}=21063928^{2}\;}$ 

${\displaystyle 43^{8}+96222^{3}=30042907^{2}\;}$ 

The first of these (1^m + 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 (a^m, bⁿ, c^k).

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.^{[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.

See also

• Sums of powers, a list of related conjectures and theorems

References

1. 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.
2. Darmon, H.; Granville, A. (1995). "On the equations z^m = F(x, y) and Ax^p + By^q = Cz^r". 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. Vol. 98. Basel: Springer. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. MR 3298238.

External links

• Perfect Powers: Pillai's works and their developments. Waldschmidt, M.