Ben Weiss
  • Member for 12 years, 2 months
  • Last seen more than a month ago
Magic trick based on deep mathematics
68 votes

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 ...

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What are the worst notations, in your opinion?
28 votes

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) = ...

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How do you become a good listener?
18 votes

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. ...

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Are there any books that take a 'theorems as problems' approach?
17 votes

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

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Cool problems to impress students with group theory
13 votes

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 ...

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Analytic density of the set of primes starting with 1
Accepted answer
12 votes

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 ...

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non-Dedekind Domain in which every ideal is generated by at most two elements
11 votes

[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 ...

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Teaching statements for math jobs?
11 votes

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....

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Set of vectors separated by at least a specified angle
10 votes

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 ...

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Why is 12 the smallest weight for which a cusp forms exists
9 votes

"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'...

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English reference for a result of Kronecker?
8 votes

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

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What are some good resources for mathematical translation?
8 votes

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 ...

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Applications of the Chinese remainder theorem
5 votes

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,...

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Books you would like to see translated into English
5 votes

Bombieri's "Le Grand Crible dans la Théorie Analytique des Nombres"

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probability in number theory
5 votes

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 ...

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Haar measure on a quotient, References for.
4 votes

"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 ...

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Interpolation of sequences by analytic functions
4 votes

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 ...

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Fundamental Examples
4 votes

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 ...

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Alternative proof of unique factorization for ideals in a Dedekind ring
3 votes

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 ...

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Factorials in Pascals Triangle
3 votes

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 ...

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Generalization of primitive roots
2 votes

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 ...

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Sumset achieving extreme upper bound
Accepted answer
2 votes

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 ...

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Bounds on solutions to Diophantine equations of the form p(x)=q(y)
2 votes

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 ...

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on prime numbers which are primitive roots of a prime
Accepted answer
2 votes

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 ...

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Prime divisors of numbers 2^n + 3
2 votes

(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 ...

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Gaps in nx (mod 1)
2 votes

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 $k\alpha (\mod 1)$, ...

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When does the zeta function take on integer values?
2 votes

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 ...

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a question on function fields (extending my previous question)
2 votes

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 ...

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Solving "a, b, a+b have given divisors" problem
2 votes

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 ...

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Bertrand's postulate
1 votes

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}{...

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