# Tagged Questions

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### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...
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### Permanent of a matrix of odd integers

It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...
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### Algebraic integers in skew fields

Hi everyone, let $D$ be a skew field, which is finite dimensional over its center $k$. Assume that $k$ is a number field, and let $\mathcal{O}_D$ be the set of elements $z\in D$ which are roots of a ...
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### How to interpret this class of numbers?

Let's say $f(p)$ is a number defined as shown below: $\hspace {10 mm}f(p) = {\sqrt {2} ^ {{\sqrt {2} ^ \sqrt {2}} ^ {... \hspace {1 mm} p\hspace {1 mm} times } }}$ What I understand is: We ...
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### When does a a rational function have infinitely many integer values for integer inputs?

Consider rational functions $F(x)=P(x)/Q(x)$ with $P(x),Q(x) \in \mathbb{Z}[x]$. I'd like to know when I can expect $F(k) \in \mathbb{Z}$ for infinitely many positive integers $k$. Of course this ...
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### Strong Bezout's Identity?

Let $\{ a_i \}_{i=1}^N$ be a set of elements of the ring of integers, $\mathbb{Z}_D$ and define $g = \text{gcd}(a_1, a_2,\ldots, a_N, D)$. Then Bezout's Identity states that there exists another set ...
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### Sum Equals Product

Sum Equal Product There are many articles in re this issue on the Web, although many are restricted to special cases (e.g., John Cook’s sum of tangents = product of tangents), and even some involving ...
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### Arctangents and the golden ratio

Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$  = \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + ...
### Proving non-existence of solutions to $3^n-2^m=t$ without using congruences
How can one construct families of cocompact discrete subgroups of the topological group $\text{SL}_2(\mathbb{C})$? Here quaternion algebra's might help, I believe, but I have some difficulties with ...