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Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind. The …

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with …

Compared to Fermat's two squares theorem, or Fermat's four squares theorem, Fermat's Little theorem is indeed Little. Not to mention the hard-to-prove Fermat Last Theorem, which goes under FLT; so th …

It is apparent that the abc conjecture is deeply related to Arakelov theory. In one direction, it is shown in S. Lang, "Introduction to Arakelov Theory", that a certain height inequality in Arakelov t …

This is a speculation and perhaps naive. The theorem of Siegel that There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is …

The article I have checked for Baker's theorem is Waldschmidt's. But the article and the citations therein are from the time of '88. Question: What is the the strongest known lower bound for Baker …

In number theory, base change of a scheme or a variety is with respect to the underlying ring or field, is viewing the same scheme/variety over an extended ring or field, but with the "same" set of eq …

I do not know about algebraic number approximations; but the most canonical approximation to real numbers by rational numbers is the continued fraction expansion. They and their convergents (of first …

First case: Complex numbers. Over $\mathbb C$ the structure as an abstract group is $\mathbb S^1 \oplus \mathbb S^1$ where $\mathbb S^1$ is the circle, i.e., $\mathbb R/\mathbb Z$. This follows as Rob …

This answer is tangential in the sense that it is speaking of the existence of solutions rather than counting them all. But I rather suspect that you would find this interesting. Suppose you have a q …

Try Serre, Topics in Galois Theory.

As noted earlier, I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly rigorous and cle …

When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results. We talk of comp …

Your question about one book for number theory is like a non-mathematician asking about one book for all mathematics. It is simply not possible. It is a growing subject in various directions. The best …

Wikipedia says that the theorem appears in Fibonacci's Liber Abaci (1202). So that could be the first European instance where this name is used(though wikipedia does not say anything about what name w …