Are there infinitely many perfect powers in the Fibonacci sequence?

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Use indices $n\in\{0,1,2,\ldots\}$ and define the Fibonacci numbers by \[ F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n. \] A nontrivial positive perfect power means an integer $a^b$ with integers $a\ge2$ and $b\ge2$. This is the convention $a,b>1$ in the catalogue's statement. Under this convention $1$ itself is excluded, even though it can be written $1^b$.

Statement recorded in the supplied source.

Are there any nontrivial positive perfect powers in the Fibonacci sequence other than $8$ and $144$? Precisely, is it true that \[ \forall n\ge0\ \forall a,b\in\mathbb Z_{\ge2},\qquad F_n=a^b\ \Longrightarrow\ n\in\{6,12\}? \] Here $\mathbb Z_{\ge2}=\{2,3,\ldots\}$, and $F_6=8=2^3$, $F_{12}=144=12^2$. If bases $0$ and $1$ are also admitted, the corresponding classification is $F_n\in\{0,1,8,144\}$. [S2] The classification was proved by Bugeaud, Mignotte and Siksek; it also gives a negative answer to the title's question about infinitely many such values. [S2]

Short English statement

Determine whether 8 and 144 are the only Fibonacci numbers that are powers with both base and exponent greater than one. The complete classification is already known; the exceptional values 0 and 1 appear only under the broader convention allowing those bases.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 27 (NT-008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Yann Bugeaud, Maurice Mignotte and Samir Siksek, Classical and Modular Approaches to Exponential Diophantine Equations I. Fibonacci and Lucas Perfect Powers, Annals of Mathematics 163 (2006), 969--1018; arXiv:math/0403046, Theorem 1, page 2. https://arxiv.org/pdf/math/0403046.