Fermat–Catalan conjecture

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

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

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), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g. ${\displaystyle 7^{5}+393^{3}=7792^{2}}$.

Known solutions

As of 2024 the following ten solutions to equation (1) which meet the criteria of equation (2) are known:^[1][2][3]

${\displaystyle 1^{m}+2^{3}=3^{2}\;}$ (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.^{[4][5]: 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.^[6]

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.

Poonen et al.^[7][8] list exponent triples where the solutions have been determined:^{[note 1]}   {2,3,7},^[8]   {2,3,8},^[9][10]   {2,3,9},^[11]   {2,2q,3} for prime 7<q<1000 with q≠31,^[12]   {2,4,5},^[10]   {2,4,6},^[9]   (2,4,7),   (2,4,q) for prime q≥211,^[13]   (2,n,4),^[14][15]   {2,n,n},^[16]   {3,3,4},^[17]   {3,3,5},^[17]   {3,3,q} for 17≤q≤10000,^[18]   {3,n,n},^[16]   {2n,2n,5},^[19]   {n,n,n}.^[20][21]   For each of these exponent triples, if there is some solution at all, it is listed among those in section § Known solutions.

Sikora partially used the cluster computers at the Center for Computational Research at University at Buffalo to test all tuples (a,b,c,m,n,k) such that min(m,n,k) ≤ 113 and a^m, bⁿ, c^k < M_min(m,n,k), where M₂ = 2⁷¹, M₃ = 2⁸⁰, M₄ = 2¹⁰⁰, and M₅ = ... = M₁₁₃ = 2¹¹³. He did not find any other solution than those above.^{[note 2][1]}

See also

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

Notes

1. The notation "{p,q,r}" means that the solutions have been determined for every permutation of (p,q,r).
2. For example, the five known large solutions were all reproduced during the test for min(m,n,k)=2, where a^m, bⁿ, and c^k were considered up to 2⁷¹.