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First, I'd like to revisit Will Sawin's upper bound, but I make no promises about off-by-one errors (which I think I made in comments elsewhere on this page). The product of any sequence of $x$ conse …

As others have mentioned, the exact sequence is continuously split. Furthermore, any lift of Frobenius will act as you expect on the subfield $\mathbb{Q}_p^{nr}$, i.e., the extension you get by adjoi …

The Ramanujan behavior is typically explained by the fact that the imaginary quadratic field $\mathbb{Q}(\sqrt{-163})$ has class number one (together with an integrality property of the j-function - s …

This is a bookkeeping post, since the answer seems to have been resolved in the comments. Somebody please vote this up once so this question leaves the "unanswered" queue. Shenghao's answer is essen …

If w is a primitive 2n-th root, then the answer is "none". If w is not primitive, then Q(w) has a square root of w if and only if and odd power of w is 1.

If you just want a classical picture over the complex numbers, the objects lying over the cusp points are Néron polygons equipped with some extra structure. To make a Néron $n$-gon, you take $\mathbb …

I don't feel particularly qualified to answer this question, but I should mention that primes are expected to be "random modulo local obstructions". By this, I mean that aside from properties derived …

This revision (June 2015) is mostly to say that I've verified the answer by ell. I found the Proceedings volume in question in the U. Tokyo library. It is basically a bound volume of photocopies of …

We get an isomorphic problem by switching $c$ with $d$, and replacing $b$ with $-b$. Then we are considering matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ in $SL_2(\mathbb{Z})$. Passage fr …

I'm not sure what "topograph" means, but if it's just a fancy word for "pictoral description of features in a region" then the answer is yes. Send your triple $(a,b,c)$ to the element $a/c+ ib/c$ in …

The conjecture that any even number (greater than 4) is the sum of two odd primes is well-known to be open. You are asking about a special case that is also open.

This was answered in the comments, so I'll just sum up: $(p)$ is the maximal ideal in $\mathbf{Z}_p^{nr}$, so we have an isomorphism $\mathbf{Z}_p^{nr}/(p) \to \overline{\mathbf{F}_p}$. The reverse …

I agree with Gjergji Zaimi's comment above: Both "Fermat test" and "Fermat primality test" are short and descriptive.

Darij Grinberg answered the question in a comment, but I thought a more geometric interpretation might be helpful. The group of positive rational numbers under multiplication is isomorphic to an infi …

From the Wikipedia page on Bertrand's postulate, Dusart (1998) showed that for all $x > 3275$, there exists a prime between $x$ and $x \left( 1 + \frac{1}{2 \ln^2 x} \right)$. You are looking for the …