<s>[INST] 2-adic expansion of (2/3) I have been asked in an assignment to compute the 2-adic expansion of (2/3). It just doesn't seem to work for me though. In our definition of a p-adic expansion we have $x= \sum_{n=0}^{\infty}a_np^n$ with $0 \leq a_0<p$.
So I use the same method as for similar questions which seems to work fine. This involves solving each congruence $\frac{2}{3} = \sum_{n=0}^{\infty}a_np^n \ (mod \ p^N)$ for $N=1,2,3,4,5$ and then we are supposed to guess the pattern and show it by summing the series.
Doing this, I get $\frac{2}{3} = 2 + 2^2 + 2^4 + 2^5 + ... \\ = (2+2^2)(1+2^3+2^6+...) \\ = 6(\frac{1}{1-2^3}) \\ = \frac{6}{-7}.$ 
Which is just completely wrong?
An example of me solving a congruence:
$\frac{2}{3} \equiv a_0 \ (mod \ 2)$ so $0 \equiv 2 \equiv 3a_0 \ (mod \ 2)$ so $a_0 = 0$.
 [/INST] I get $2^1+2^2+2^4+2^6+\ldots$, with no $2^5$.  In the $2^4$ place, you have $3\times1+1=4=0$ carry $2$, so you need $0\times 2^5$ to give $0$ carry $1$
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