Let $F_k$ be the $k$-th Fibonacci number. Cassini's identity is $F_{k-1}F_{k+1}-F_k^2=(-1)^k$, hence $F_{k-1}^2+F_{k-1}F_k-F_k^2=(-1)^k$. Raising this to the third power and reordering terms yields
\begin{equation}
2(F_{k-1}^6-F_k^6)+(F_k^2+F_{k-1}F_k-F_{k-1}^2)^3=(-1)^k,
\end{equation}
which explains infinitely many pairwise coprime integer solutions.

See also sequence [A337929][1] at the OEIS.


  [1]: https://oeis.org/A337929