Are Irrational Square Roots Normal?

Open Question

It is not known whether irrational square roots are normal.

Progress

It has been established by testing the distribution of the digits of the square roots of the integers $2$ to $15$, excluding $4$ and $9$, in bases $2$, $4$, $8$ and $16$, that they may well be normal.

Sources

• 1970: W.A. Beyer, N. Metropolis and J.R. Neergaard: Statistical Study of Digits of Some Square Roots of Integers in Various Bases (Math. Comp. Vol. 24: pp. 455 – 473)  www.jstor.org/stable/2004493

Beyer is mistakenly reported as Beyler in David Wells: Curious and Interesting Numbers.

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 12345 67891 01112 13141 51617 18192 02122 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$