User:Dominus/Octopuses

Frequently Asked Questions about Octopuses

Q: How many arms do octopuses have?

A: As the name suggests, an octopus in ${\mathbb {R} }^{3}$ has eight arms. In general, an octopus in ${\mathbb {R} }^{n}$ has $2^{n}$ arms. The existence of octopuses with fractal Hausdorff dimension was an open conjecture until 1983, when Miller and Pine constructed a 10-armed octopus. The construction generalizes to any number of arms that of the form 2p for prime p. The other cases remain open, although the the existence of an octopus with an odd number of arms greater than 1 was recently shown to be equivalent to the ABC conjecture.