If every $0$ digit in the expansion of $\sqrt{2}$ in base $10$ is replaced with $1$, is the resulting sequence eventually periodic?

If every $0$ digit in the expansion of $\sqrt{2}$ in base $10$ is replaced with $1$, is the resulting sequence eventually periodic?

• Since the sequence is not periodic, if you change zero to any number, it will still be non periodic since other numbers at other places also decide periodicity. Nov 1, 2016 at 9:34
• @jnyan that's not true Nov 1, 2016 at 9:55
• Possibly relevant : math.stackexchange.com/questions/971137. I quote Milo Brandt : "Strictly speaking, it is unknown whether $\pi$ with every $4$ changed to $6$ would be irrational (since we don't know even that every digit of $\pi$ occurs infinitely often), but if it's a normal number, then it would still be irrational". Nov 1, 2016 at 17:12
• @jnyan : I think you actually proved the converse : if $x$ is rational, then replacing every $0$ digit by $1$ yields a rational number $y(x)$. But here it is a different question: $x_0=\sqrt 2$ is irrational, but does it follow that $y(x_0)$ is irrational? For instance, $x=0.10100100010000…$ is irrational but $y(x)=0.1111111…$ is rational. Nov 1, 2016 at 17:13
• @jnyan : the question is about changing "every $0$ digit", not only one of them. Nov 1, 2016 at 17:20