# Is there any kind of known pattern to $\sqrt 2$ in base 2?

Is there any kind of known pattern to $\sqrt 2$ in base 2? Is there any classification categories for decimal digits of numbers that for example would put $\sqrt 2, \sqrt 3 \cdots \sqrt n$ into separate category than $\sqrt[3] 2, \sqrt[3] 3 \cdots \sqrt[3] n$

Or given a decimal expansion with arbitrary precision, or definition of digit in the nth place (not necessarily decimal expansion) to get a probability of which class of numbers the number is likely to be? e.g. transcedental or algebraic of degree k?

• If there was one on base $2$, then there would be one on base $10$ as well, making $\sqrt2$ rational. Nov 9, 2015 at 9:47
• @barakmanos : The question did not ask for a "repeating pattern", e.g. liouville's number has a simple pattern and it is transcendental. Nov 9, 2015 at 9:50
• I believe it remains unknown whether $\sqrt {2}$ is "normal" in any base i.e. for each natural number $n$ having uniform distribution of all $n$-tuples of digits in that base. Nov 9, 2015 at 10:00
• The wikipedia page is pretty interesting, and points out that it is unknown if $\sqrt 2$ is normal. In fact, we have never proven that a number is normal, except for numbers which were constructed to be normal. en.wikipedia.org/wiki/Normal_number Nov 9, 2015 at 10:14
• Apr 17, 2017 at 8:03

One expression for the square root of 2 is given by $$\sqrt{2} = \sum_{k=0}^{\infty} (2k+1) \binom{2k}{k} 2^{-3k -1}$$ which is rather useful for computing the digits exactly. It also means that any pattern in the binary digits of $$\sqrt{2}$$ should be somewhat related to the the digits of the central binomial coefficients. There's probably a combinatorial interpretation of this.
Incidentally, $$(2k+1) \binom{2k}{k}$$ is asymptotic to $$\frac{2^{2k + 1}}{\sqrt{\pi}}$$ so the number of terms affecting the kth digit is linear in k. Also, the greatest power of 2 dividing the central binomial coefficient is known as Gould's sequence, i,e, $$\binom{2k}{k} = 2^{2^{h(k)}}$$ where $$h(k)$$ is the hamming weight of k.