I'm trying to find represent expansion of $a^{-1}$, if $$a = a_{r} \cdot p^{r} + a_{r+1} \cdot p^{r+1} +...$$
I have tried get solution supposing that $a \cdot a^{-1} = 1$, but stucked. I don't even see any pattern, for example with $p = 3$:
$$3 = 1 \cdot 3^{1} $$
$$\frac{1}{3} = 1 \cdot 3^{-1}$$
While
$$2 = 2 \cdot 3^{0}$$ $$\frac{1}{2} = 2 \cdot 3^{0} + 1 \cdot 3^{1} + 1 \cdot 3^{2} + ...$$
And
$$-3 = 2 \cdot 3^{1} + 2 \cdot 3^{2} + 2 \cdot 3^{3} + ... $$ $$-\frac{1}{3} = 2 \cdot 3^{-1} + 2 \cdot 3^{0} + 2 \cdot 3^{1}$$
Thanks for any help.