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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5
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1answer
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+50

What is known about these arithmetical functions?

Let $n=\prod_p p^{c_p}$, $N\in \mathbb N$ and $$ \alpha_N(n)=\prod_p p^{c_p \bmod N}. $$ The function $\alpha_N$ is multiplicative since $\alpha_N(n)\alpha_N(m)=\alpha_N(nm)$ for co-prime $n$ and $m$ ...
1
vote
0answers
47 views
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Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
17
votes
0answers
226 views
+100

Meromorphic functions on $U^2 = T^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
4
votes
1answer
57 views
+100

Extending 2-adic valuation to real numbers

When proving Monsky's theorem, one of the steps, which, from what I have so far seen, no proof can avoid, is extending the 2-adic valuation to all real numbers, so that it still satisfies ...
7
votes
2answers
74 views
+50

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...