Freeman Dyson has claimed that
$\not \exists m,n Reversed(2^n) = m ^ 5 $
(where Reversed(l) just is the reverse of the digits of l in base 10), is probably an example of an unproveable truth (source), and that even if it's not, there are many similar examples, some of which will be examples. As a heuristic argument, he says that a proof would have to rely on some pattern in the digits of powers of two, but those seem to be random.
Is this heuristic argument reasonable? I was surprised because I had thought that a search for undecidable arithmetic statements in Peano Arithmetic (let alone ZFC or something stronger) was rather challenging. But maybe that's only for (a) provably unprovable statements or (b) interesting statements. I'd tend to assume Dyson knows what he's talking about here, but am still curious.