Fermat–Catalan conjecture

Template:Short description Template:Use dmy dates 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 Template:NumBlk 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 Template:NumBlk 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. $7^5+393^3=7792^2$.

Known solutions

As of 2024, the following ten solutions to equation (1) which meet the criteria of equation (2) are known:

$1^m+2^3=3^2$ (for $m>6$ to satisfy Eq. 2)

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

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

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

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

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

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

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

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

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

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 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' 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.Template:Rp 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.

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. list exponent triples where the solutions have been determined:   {2,3,7},   {2,3,8},   {2,3,9},   {2,2q,3} for prime 7<q<1000 with q≠31,   {2,4,5},   {2,4,6},   (2,4,7),   (2,4,q) for prime q≥211,   (2,n,4),   {2,n,n},   {3,3,4},   {3,3,5},   {3,3,q} for 17≤q≤10000,   {3,n,n},   {2n,2n,5},   {n,n,n}.   For each of these exponent triples, if there is some solution at all, it is listed among those in section Template:Section link.

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.

See also

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

External links

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