[sweezyjeezy]

Base‑10 is just our chosen way of writing numbers, it doesn’t need to have any deep relationship with the arithmetic properties of sequences like the powers of 2. For most series (Fibonacci numbers, factorials etc), the digits for large members will be essentially random, their digits don't obey any pattern - it's just two unconnected things. It seems extremely likely that 2048 is the highest, but there might not be a good reason that could lead to a proof - it's just that larger and larger random numbers have less and less chance of satisfying the condition (with a tiny probability that they do, meaning we can't prove it).

Interestingly, there are results in the other kind of direction. Fields medalist James Maynard had an amazing result that there are infinitely many primes that have no 7s (or any other digit) in their decimal expansion. This actually _exploits_ the fact that there is no strong interaction between digits and primes - to show that they must exist with some density. That kind of approach can't work for finiteness though.

[kens]

Yes, I find math problems that depend on base 10 to be unsatisfying because they rely on arbitrary cultural factors of how we represent numbers. "Real" mathematics should be universal, rather than just solving a puzzle.

Of course, such a problem could yield deep insight into number theory blah blah blah, but it's unlikely.