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This is somewhat recreational, so excuse me for the lack of rigorosity. Say we expand $2^n$ for given $n$ in $base 10$. Then, there will be a ratio of even/uneven digits in the expansion. (i.e. $2^{30}$$=1073741824$, which has five uneven digits and five even digits.)

Now what happens with this ratio when $n$ goes to infinity? Are we allowed to make a statement about it? And if so, is there a proof?

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    In short: (a) the answer is probably that the ratio goes to half-and-half, (b) this probably holds true in most non-power-of-two even bases, and (c) it's surely almost impossible to prove. – 2017-01-23
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    (Also, I count five odd digits and five even digits in your example?) – 2017-01-23
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    $0$ is an even number. See [this question](http://math.stackexchange.com/q/15556/1508). – 2017-01-23
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    @steven I cant count lmao. Why would it be almost impossible to prove? – 2017-01-23
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    One thing that is known: if $a(n)$ is the number of odd digits in $2^n$, $\sum_{n=1}^\infty a(n)/2^n = 1/9$. See [OEIS sequence A055254](https://oeis.org/A055254) – 2017-01-23
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    @RobertIsrael: I thought you made a mistake, but in fact the mistake is in OEIS: the sum given in the FORMULA section should be $10/9$, not $11/9$. – 2017-01-23
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    Almost impossible because we know so little about how expansions of numbers in different bases interact with each other besides the trivial. – 2017-01-23
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    well, we know that the distribution of the first digit of the powers eventually follows Benford's law, and that the last digit is always even except for 1 :-) I think that we can find formulas for second, third... digit and that it could be shown that the total ratio tends to 1:1 – 2017-01-23
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    @mau I agree with Steven. I do not see a chance to prove it. – 2017-01-23
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    $82$ upvotes for the question "Is zero odd or even". How can $0$ be an odd number ? The definition is : A number $n$ is called even, if $n\equiv 0\mod 2$ and odd, if $n\equiv 1\mod 2$. I have not looked at the question, but I wonder in which sense $0$ can be considered to be odd. – 2017-01-23
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    @TonyK OEIS is a nice database for sequences, but contains unfortunately many errors (although most sequences are marked "approved"). – 2017-01-23
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    @Peter: It's easy to see why people might be uncertain. Many children are taught at an early age that the odd numbers are $1,3,5,\ldots$ and the even numbers are $2,4,6,\ldots$ Nobody thinks that $0$ is odd; they just think it is uncategorised. – 2017-01-23
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    Im still not satisfied with the answers. I still don't understand why you people give so little chance of proving it. – 2017-01-23

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