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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17
votes
2answers
289 views
+50

Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$.

Prove that $\mathbb{Q}(\sqrt{-11})$ is of class number $1$. I have found that the ideal $(2)$ of the integer ring $\mathbb{Z}[(1 + \sqrt{-11})/2]$ of $\mathbb{Q}(\sqrt{-11})$ is a prime ideal. ...
11
votes
1answer
254 views
+50

Numbers having in decimal representation no common digits with all their proper divisors

Let us call a positive integer having in decimal representation no common digits with all its proper divisors "a good number". $54$ is a good number : $1,2,3,6,18,27$ $48$ is not a good number : ...
16
votes
2answers
267 views
+50

Geometric mean of prime gaps?

The arithmetic mean of prime gaps around $x$ is $\ln(x)$. What is the geometric mean of prime gaps around $x$ ? Does that strongly depend on the conjectures about the smallest and largest gap such as ...
9
votes
0answers
79 views
+100

Meromorphic functions on $Y^2 = X^3 + 1$, genus.

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