In mathematics the $p$-adic number system for any prime number $p$ extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

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### Finding an example for an elliptic curve over the p-adics with bad reduction but potential good reduction

Problem I would like to find an elliptic curve $E$ over $\mathbb{Q}_p$ given by the equation $$E: \quad y^2 = x^3 - 27 c_4 x - 54 c_6.$$ with the following properties: $E$ does not have good ...
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### A question on Fontaine's periods rings

Let $K$ be a complete discrete valued field of characteristic zero with perfect residue field $k$ of characteristic $p>0$, $\mathcal{O}_K$ its ring of integers, $C$ the completion of an algebraic ...
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### In p-adic metric, what is the distance between 0.9999… and 1? [closed]

In 10-adic (though 10 is not a prime number) metric, we know that $\Vert1-0.9999...\Vert=\Vert\lim_{n\rightarrow \infty}\frac{1}{10^n}\Vert=\lim_{n\rightarrow \infty}10^n\rightarrow\infty$. ...
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### Computing ramification in extension of complete DVRs

Assume I am given a finite primitive extension of complete discretely valued fields $L=K(\alpha)/K$, say with monic integral minimal polynomial $f$ for $\alpha$. How does one systematically compute ...
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### Proof explaination - $\sum_{i=1}^{n} \frac{1}{i}$ is not an integer for $n>1$

I was reading a proof to the following fact: for $n>1$, $\sum_{i=1}^{n} \frac{1}{i} \notin \mathbb{Z}$. The proof is as follows: Denote for prime $p$ by $v_p(a)$ the p-adic valuation of $a$. Write ...
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### Proof that (x OR c) = x + c - (x AND c)

Proof that (x OR c) = x + c - (x AND c), where $x$ -- p-adic number in $Z_2$ and $c$ -- is positive integer. AND, OR - bitwise operations
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### Logarithm and Lubin-Tate formal group

Let $K$ be a finite extension, by Milne's online note "class field theory", $m_{\mathbb{C}_p}$ has a natural $O_K$ module structure where the action is given by $[a]_f$. For such a $f$, there exists a ...
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### What is the measure of the function?

Is the $T$-function $f(x) = (x \text{AND} c) + (x^2 \text{OR} c)$, where $c$ is positive integer ergodic in the space $Z_2$ (p-adic numbers)? What is the measure of this function? I am trying to use ...
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### Showing that Q is not complete with respect to the 2-adic and 3-adic absolute value

I have seen here how to show that Q is not complete with respect to the $p$-adic absolute value, where $p\geq5$. Is there a similar proof/idea for $p=2$ and $p=3$?
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### Need help in an argument related to P-adic valuation

I am unable to understand why an argument related only to p-adic number theory must be true . Question: Assume (2.5) equivalent to equation S= -P to simplify notation( Here S and P are sums ...
### $3^n$ does not divide $4^n+5$ for $n\geq 2$
Question as in the title : does anyone know how to prove that $3^n$ does not divide $4^n+5$ for $n\geq 2$ or find a counterexample ? My thoughts : (1) I have checked that this is true for $n\leq 1000$...