This was fascinating for me. Somehow the man takes a bagel and with one cut arrives with two pieces that are interlocked. Whether this qualifies as "magic" I dunno (it's hard to say once the trick's ...

As Trevor Wooley used to always say in class, ``Vinogradov's notation sucks....the constants away." For those who don't know, Vinogradov's notation in this context is $f(x)\ll g(x)$ meaning $f(x) = ...

The best advice I ever received was from Jordan Ellenberg, I hope he doesn't mind my rephrasing of it here. When sitting in a seminar, try to figure out an interesting question to ask the speaker. ...

My favorite such book is Problems in Analytic Number Theory by Ram Murty. There could not be enough good things said about it.

This is related to many of the answers already here, but a little different. When I was an undergrad, I got fixated on the following problem: Given a deck of $n$ cards, how many perfect shuffles does ...

I think instead of posting my own explanation (which will only lose something in the translation) I'll instead refer you to two very interesting papers (thanks for posting this question, I haven't ...

[Edited to restrict to the case of quadratic orders. --PLC] Take any non-maximal order of a quadratic number field. This is not Dedekind because it fails to be integrally closed in its field of ...

Just to expand on Michael's last comment, I found that same advice article, and think it's worth posting a link. In particular, it was the only thing I used as a guide towards writing my own statement....

The subject name you are looking for is spherical codes. A good reference for this subject is Conway and Sloane's "Sphere Packings, Lattices, and Groups." In chapter 9 they give the details of the ...

"Lattice Polygons and the Number 12" by Poonen and Rodriguez-Villegas relates the 12 in the Riemann-Roch theorem, to the 12 in the weight of the cusp form, to a property of convex lattice polygons. It'...

Bombieri and Gluber's recent book "Heights in Diophantine Geometry" has a proof of this in chapter 1.

I've found that Google has a translator application which is wonderful. If you type in a sentence, it does the correct translation; it has even known mathematical terms when presented in the correct ...

One answer I don't see here: Lagrange interpolation. If one takes, for example, the ring $\mathbb{Q}[x]$, and realizes CRT as a statement about rings and direct sums of $R/P$ over a set of co-prime $P,...

The one I learned from is Tenenbaum. My personal favorite application is to derive heuristics for the twin prime conjecture (and more general Hardy-Littlewood conjecture). For an excellent ...

"Fourier Analysis on Number Fields" by Ramakrishnan and Valenza deals with many of the same topics, but starts in chapter 1 with exactly this material and works up to Tate's thesis in chapter 7. I ...

There is a classic result of Ramanujan known as his Master Formula which Wolfram has here: http://mathworld.wolfram.com/RamanujansInterpolationFormula.html To summarize briefly (and coarsely): if ...

I think the question of whether or not $li(x) := \int_{2}^{x} \frac{dt}{\ln(t)}$ was contained in $\mathbb{C}\left(x, \ln\left(x\right)\right)$ spawned the field of galois theory of differential ...

I believe that the proof in Marcus's "Number Fields" contains a proof which does not rely on integral closure except to say that if an element satisfies a monic polynomial, it is in the domain. I'll ...

This is only a start, but maybe it'll suggest to others how to proceed. If $p$ is a prime number, then there is no integer $1 \le n < p$ and integer $m$ with $${p\choose{n}} = m!$$ This is because ...

There are good results related to what you are asking. Hans Roskam stated and proved a quadratic analogue of Artin's conjecture (and I think his paper is much easier to read than Hooley's). He studies ...

Elaborating on what Gerhard Paseman points out, selecting sets $A_i = \{kh^i \mid 0 \le k < h\}$ will work. More generally, if you define your sets inductively (and restrict to only using positive ...

I don't know exactly what the problem you're working on is, you want a bound in the case that it is finite? Or you want to prove that there are finitely many solutions? The papers I'd recommend are ...

Gupta and R. Murta showed that there were infinitely many primes $p$ for which there are infinitely many $q$. Heath-Brown generalized this to show that there are at most two prime numbers $p$ for ...

(Edited as the comments below suggest) The ABC conjecture seemed to me like it would play a roll, however it comes up a little short: "Are there infinitely many primes $p$ so that for each $p$ there ...

This reminds me of the following cute statement (which I believe has been recently generalized to arbitrary manifolds, but I couldn't find the paper): Given the sequence of points $k\alpha (\mod 1)$, ...

Let's look at $\zeta(s)$ for some large $\sigma$ (the real part of $s$). We can bound the function by $\int_1^\infty \frac{dx}{x^\sigma} + 1$, which is $\frac{\sigma}{\sigma-1}.$ So all these values ...

You will have more points than just the ones listed. To start, notice that $x$ and $y$ are both invertible. This is because $1 - a^5$ is invertible, and this equals $x^5$. This lets you write $1/x$ as ...

This sounds to me like it is related to some recent work of Lagarias and Soundararajan: http://arxiv.org/pdf/0911.4147 Their paper has some relations to the ABC conjecture and to GRH. Hope this ...

One remark to relate Bertrand's postulate to the prime number theorem: Chebyshev's work was related to bounding ratios of factorials--in particular $\frac{2n!}{n!n!}.$ His later proof that $C\frac{x}{...