# Are there more than 2 digits that occur infinitely often in the decimal expansion of $\sqrt{2}$?

The other day I got to thinking about the decimal expansion of $\sqrt{2}$, and I stumbled upon a somewhat embarrassing problem.

There cannot be only one digit that occurs infinitely often in the decimal expansion of $\sqrt{2}$, because otherwise it would be rational (e.g. $\sqrt{2} = 1.41421356237\ldots 11111111\ldots$ is not possible).

So there must be at least two digits that occur infinitely often, but are there more? Is it possible that e.g. $\sqrt{2} = 1.41421356237\ldots 12112111211112\ldots$?

• I think that every algebraic irrational number contains each one of the digits (on any natural base!!!) infinitely many times. – barak manos Dec 15 '15 at 13:45
• If there is a "general" first part $g$ of the digit expansion of $\sqrt 2$ which contains somehow all digits (but has only finitely many digits (let its number be called "m") and is thus rational) and then a "special" second part of infinite length which contains, say only the digits $1$ and $2$ then we would have $\sqrt 2 = g + 10^{-m} s$ and $2 = g^2 + 2 gs + s^2$ and I would try then to find configurations in $s$ which disallow consecutive zeros at the end. – Gottfried Helms Dec 16 '15 at 10:58

• I think this is a bit quick-shot. Recently (ok: "recently") Plouffe and the Borweins found that algorithm which produces the digits of $\pi$ (in hex-base) and by such a formula one should be able to say something about frequencies. I don't know actually about the same in dec-base however... – Gottfried Helms Dec 16 '15 at 10:52
• @GottfriedHelms I assume this is the one: en.wikipedia.org/wiki/… I haven't seen that before, but maybe you are right that it is possible to figure out when the algorithm returns a certain digit, and if you can do this for all digits, then conclude that the hex-expansion of $\pi$ contains all digits infinitely often. Maybe you can even say something about the frequency of each. I just checked, and it looks like though that $\pi$ is still not known to be normal in any base. – Mankind Dec 16 '15 at 11:15