Do the decimal digits of $\pi$ and $\sqrt{2}$ coincide infinitely often?

Question

I suspect the answer to the title question is "yes," but can it be proved? One would expect digits match in one tenth of cases.

Answer 1

I'm sure your expectation is correct in reality; but as commenters have mentioned, this is the type of problem that is currently hopeless given what we know about proving such statements.

Just to point out how far away we seem to be from being able to address this problem, let me make the following provocative and surely false assertion:

After finitely many digits, the decimal expansion of $\pi$ consists only of $3$s and $8$s and the decimal expansion of $\sqrt2$ consists only of $1$s and $5$s.

That statement is ridiculous—but we can't even disprove that, as far as I'm aware! And the conjecture in the OP is far stronger than simply disproving the above provocative statement.