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You will end up taking m as around log n/loglog n, asymptotically, by the Prime Number Theorem? So again, your motivation. For small n something less easy to describe but computable will occur. For la …

As you probably know, you are also asking for a particular Gaussian prime ideal $(\pi )$ with norm $p$. Which is to say that such a choice of square root of $-1$ is the same as giving a homomorphism f …

A simple reason: this is a function of $n$ satisfying significant congruences. If it vanishes, that is further congruence information.

The motivation "because I'd like to know [...] if it is impossible to prove FLT using elementary methods" seems to require comment. It is much more likely (in my view) that it is true that FLT can be …

You can evaluate a series for cotangent at the square root of -1: see http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions .

If I'm not mistaken, the slightly stronger result is true, that the lcm of orders of Frobenius elements must be even (forget ramified primes, that is). Isn't that a corollary of the Chebotaryov densit …

Your condition (2) shows that you are looking for integral points (m, n) on a certain curve of genus 1. These are probably already rare. There is a side condition on n which you also require. I expect …

If you take any vector in Euclidean space, and look at its integer multiples modulo the integer lattice, we know what happens qualitatively. They are dense in a certain subtorus of the obvious torus. …

I suggest a course of treatment with Mordell's book Diophantine Equations. It is perhaps an extreme case, but it represents number theory "before Bourbaki got his hands on it", as a leading British ma …

I can't help thinking that the answer ought to come out of a combination of standard techniques. First, indeed, reduce to fractional parts: but with a caveat, that it would be sensible to do some mani …

Why not try to "solve a harder problem", and produce Eisenstein polynomials of whatever degree you want, over any number field? These give totally ramified extensions.

The general phenomenon, of ideals becoming principal in an extension, is called "capitulation". There has been work going on for a century now. Some results are mentioned in this grant report I found: …

I'd lay odds on a counterexample. This is going to be a bit vague, but I assume the reason you're looking at a coset is down to Pontryagin duality at the local field level. That it is the coset for a …

Yes. The determination of Gauss sums with characters of prime order at least 5 is an unsolved problem.

It looks to me like the intended method is to use the loglog divergence of the sum of the reciprocals of the primes on the first sum, and then presumably the Prime Number Theorem with error term on th …