MHB What else could we do? (p-adic expansion)

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The discussion focuses on finding the p-adic expansions of 1/p and 1/p^r in the field Q_p. It is noted that solving the congruences px ≡ 1 (mod p^n) and p^r x ≡ 1 (mod p^n) does not yield solutions, as these expressions do not represent p-adic integers. The correct p-adic expansions are identified: for 1/p, the expansion has all digits zero except for a-1 = 1, and for 1/p^r, all digits are zero except for a-r = 1. Examples provided include the 5-adic representations of 1/5 and 1/5^3. The discussion concludes with a clarification of the conventions used in these expansions.
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Hello! (Wave)

I want to find the p-adic expansion of $\frac{1}{p}$ and $\frac{1}{p^r}$ in the field $\mathbb{Q}_p$.

So, do I have to solve the congruences $px \equiv 1 \pmod {p^n}, p^r x \equiv 1 \pmod { p^n }, \forall n \in \mathbb{N} $, respectively?

But.. these congruences do not have solutions, right? (Thinking)

What else could we do, in order to find the p-adic expansion of $\frac{1}{p}$ and $\frac{1}{p^r}$ in $\mathbb{Q}_p$ ? (Worried)
 
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If you are trying to express 1/p and 1/p^r as a list of p-adic integer digits,
{a0, a1, a2, ... }
then it is correct " these congruences do not have solutions"

These are not p-adic integers. The digit list has non-zero an, for n<0.

Case 1/p:
all digits zero except a-1 = 1

Case 1/p^r:
all digits zero except a-r = 1

Examples: 1/5 and 1/5^3 as 5-adic
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Conventions as in this DEMO
 

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