# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Different Ideals and their relationship to tame/wild ramification

I am looking at Marcus' book "Number fields", precisely at exercises that lead to Hilbert's formula $k= \sum_{m\ge0} (|V_m|-1)$; that can be applied in turn to prove that, if $L/K$ is a Galois ...
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### Bounding the size of integer solutions of curves $x^{r}+y^{s}=z^{t}$

I was reading Poonen's paper "Twists of X(7) and primitive solutions of $x^{2}+y^{3}=z^{7}$ (available here) where he describes how he found all primitive solutions of the above curve. A solution is ...
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### How much countable additivity does asymptotic density have?

For $A\subseteq \mathbb N = \{1,2,3,\ldots\}$ let $d(A)$ be the "density" defined by $$d(A) = \lim_{n\to\infty} \frac{|A \cap \{1,\ldots,n\}|} n \tag 1$$ whenever that limit exists. This is ...
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### Prove this inequality for sufficiently large $n$

Prove that the above inequality holds for sufficiently large $n$: $$\pi(2n) - \frac{3}{2} \pi(n) \ge O\left(\frac{\ln n}{(\ln \ln n )^2}\right)$$ $\ln n$ denotes to natural logarithm and $\pi(n)$ is ...
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### Convergents of continued fractions

Let $d$ and $m$ be positive integers such that $d$ is not a square and such that $m\leq\sqrt{d}$. I want to prove that if $x$ and $y$ are positive integers stafisfying $x^2-dy^2=m$ then $x/y$ is a ...