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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?

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  • $\begingroup$ If there was one on base $2$, then there would be one on base $10$ as well, making $\sqrt2$ rational. $\endgroup$ Nov 9, 2015 at 9:47
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    $\begingroup$ @barakmanos : The question did not ask for a "repeating pattern", e.g. liouville's number has a simple pattern and it is transcendental. $\endgroup$
    – jimjim
    Nov 9, 2015 at 9:50
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    $\begingroup$ 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. $\endgroup$
    – Simon
    Nov 9, 2015 at 10:00
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    $\begingroup$ 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 $\endgroup$
    – pancini
    Nov 9, 2015 at 10:14
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    $\begingroup$ See this :community.wolfram.com/web/community/groups/-/m/t/1063480 $\endgroup$ Apr 17, 2017 at 8:03

2 Answers 2

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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.

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there is a method to calculate decimal places of root2, probably that could help find a pattern with it. Although this is not a very obvious one, it probably could (very loosely) help to find a pattern. The method works a bit like dividing, so you have to calculate the next decimal place, then the next. It does not work on approximations by the way.

http://www.murderousmaths.co.uk/books/sqroot.htm

On this link you will see a few methods, but scroll down past the cyan shortcut box and you will see the one shown by Frank La Fontaine.

I'm sorry that I can't show an actual pattern, but hopefully this helps.

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