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This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability. Likely my favorite fun fact in all of number theory is the jux …

Kidding, kidding. But I do have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small. Here's a result of Eichler (remark after Theorem 6.23 in Washingt …

This is a beautiful aspect of the analogy. In fact, I think it's important and cool to view it as a consequence of previously-established aspects of the analogy, rather than a new analogy-by-fiat. U …

The only proof that I know that $(\mathbb{Z}/3\mathbb{Z})^3$ does not appear as the class group of a quadratic imaginary number field is by brute force search. Roughly, the idea is that since class n …

The odd perfect number problem likely needs no introduction. Recent progress (where by recent I mean roughly the last two centuries) seems to have focused on providing restrictions on an odd perfect …

This result is pretty shy of needing the full Hasse-Minkowski Theorem. Indeed, since Fermat already knew which integers were a sum of two integer squares, it would suffice for him to show that those …

The short answer to your question is basically no, there's essentially no connection between the prime powers $q^i$ dividing $h_p$ and $h_{-p}$. It's true that there's a general relationship betwee …

Two things: 1) Yes, certainly. By class field theory and the finiteness of the class group, the maximal abelian unramified extension of any number field is of finite degree. Thus any infinite un …

In his article On the "gap'' in a theorem of Heegner, Stark does a pretty thorough job of explaining where people thought the purported gap came from, to what extent it actually was a gap, and what yo …

No, I'm pretty sure not. In general, the theory is much more developed for the maximal pro-p-quotient of the groups you're asking about, and even in this more explored setting, not a single explicit …

To add on to Pete's answer, let me comment that the differences are even more pronounced if we look at the maximal pro-$p$ quotient $\pi_1(\operatorname{Spec}(\mathbb{Z}\left[\frac{1}{p_1p_2\cdots p_r …

The question itself is certainly still open. Mostly as an exercise for myself, I'll coalesce my comments above into an answer, and add in some details about where various pieces of the philosophy com …

It is very likely that every (positive) odd number is covered by a sum of this type. As Robin Chapman points out, this is equivalent to asking, for a given odd number $h$, whether there exists an odd …

Sure. $a\equiv b\pmod{\mathfrak{P}}$ just means $a-b\in\mathfrak{P}$. Taking norms to any subfield $K$ of $\mathbb{Q}(\zeta_n)$ (e.g., $\mathbb{Q}$ or $\mathbb{Q}(\zeta_m)$) gives you $N_{\mathbb{Q} …

"Do these objects have a name" Probably not. "Multiplicative subgroup of $S^1$ generated by algebraic integers" is pretty descriptive, and not of such fundamental importance that it's worth shorteni …