TITLE: Is 8 the largest cube in the Fibonacci sequence?
QUESTION [26 upvotes]: Can you prove that 8 is the largest cube in the Fibonacci sequence?

REPLY [4 votes]: In Siegel's article Zum Beweise des Starkschen Satzes. Inventiones mathematicae (1968), Siegel reduces the class number one problem for imaginary quadratic fields to the determination of Fibonacci cubes. The latter problem he solves in an elementary way. However, he does not mention Fibonacci numbers.