Fermat–Catalan conjecture

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

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

(1)

has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (a^m, bⁿ, c^k); here a, b, c are positive coprime integers and m, n, k are positive integers satisfying

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

(2)

This inequality restriction on the exponents has the effect of precluding consideration of the known infinitude of solutions of (1) in which two of the exponents are 2 (such as Pythagorean triples).

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

$1^m+2^3=3^2\;$

$2^5+7^2=3^4\;$

$13^2+7^3=2^9\;$

$2^7+17^3=71^2\;$

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

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

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

$9262^3+15312283^2=113^7\;$

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

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

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 we can pick any m for m>6), these solutions only give a single triplet of values (a^m, bⁿ, c^k).

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;^{[2][3]:p. 64} but the full Fermat–Catalan conjecture is a much stronger statement since it allows for an infinitude of sets of exponents m, n and k.

The abc conjecture implies the Fermat–Catalan conjecture.^[1]

Beal's conjecture is true if and only if all Fermat-Catalan solutions use 2 as an exponent for some variable.

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, 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.
3. Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review 1 (1).

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