How to find a p-adic expansion?

Question

I've been reading about p-adic numbers recently and came across a question that asks to find the $5$-adic expansion of -3.

I've been unable to find any similar examples so I can see how to work my way through this problem.

Would someone be able to work through this example or a similar one to show me how to find p-adic expansions in general?

Answer 1

We always have $-1 = (p-1)+(p-1)p+(p-1)p^2 + \dotsb$ in $\mathbb Z_p$ by the geometric series. With $p=5$, we obtain

$$-3 = -1-2=2+4\cdot 5+4 \cdot 5^2 + \dotsb$$

Answer 2

One can even adopt a sort of grade-school approach. Let

$$-3=\sum_{k\ge 0}a_k5^k=(\ldots a_3a_2a_1a_0)_5\;;$$

adding $3$ must yield $0$, so $a_0$ must be $2$. That produces a carry, so $a_1+1$ must produce a $0$, and $a_1$ must be $4$. That again produces a carry, and it’s clear that we must have $a_k=4$ for all $k\ge 1$. Thus,

$$-3=(\ldots4442)_5=2+\sum_{k\ge 1}4\cdot5^k\;.$$