Most Popular
1500 questions
45
votes
6 answers
Can we color Z^+ with n colors such that a, 2a, ..., na all have different colors for all a?
For example for n=2 coloring odd numbers red, numbers of the form 4k+2 blue and so on works.
This problem was posed in the KoMaL for n+1 prime, by Peter Pach Pal. I verified it for all n<30, I think with a computerprogram one can easily verify it…
domotorp
- 18,332
- 3
- 54
- 121
45
votes
5 answers
Integer-valued factorial ratios
This historical question recalls
Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation
Chebyshev used the factorial ratio sequence
$$
u_n=\frac{(30n)!n!}{(15n)!(10n)!(6n)!}, \qquad n=0,1,2,\dots,
$$
which assumes…
Wadim Zudilin
- 13,404
45
votes
14 answers
Examples of undergraduate mathematics separation from what mathematicians should know
I'm looking for examples of four kinds of things:
Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying examinations) but that most mathematicians aren't really…
Vipul Naik
- 7,240
45
votes
8 answers
How hard is it to get tenure in mathematics?
My curiousity provoked by this question and Igor's answer, I'd like to know: how many mathematics PhD's who want to get a tenured job as a research mathematician actually get that job?
As a first approximation, I'd like to know how what percentage…
GS
45
votes
3 answers
Why such an interest in studying prime gaps?
Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions :
lower bounds (recent works by Maynard, Tao et al. [1])
upper bounds (recent works by Zhang and the whole Polymath 8…
Desiderius Severus
- 5,324
45
votes
0 answers
What is the "real" meaning of the $\hat A$ class (or the Todd class)?
In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the index theorem for Dirac operators here, where $\hat…
Sebastian Goette
- 6,666
45
votes
3 answers
Intuition behind Riemann's bilinear relations
I'm trying to understand Riemann's bilinear relations on the normalized period matrix of a Riemann surface. Recall that they say the following. Let $X$ be a compact Riemann surface of genus $g>0$. Fix a standard basis $\{a_1,b_1,\ldots,a_g,b_g\}$…
BFA
- 451
45
votes
5 answers
How many rearrangements must fail to alter the value of a sum before you conclude that none do?
This will not be altogether unrelated to this earlier question.
For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of real numbers, if $\displaystyle f\mapsto \lim_{n…
Michael Hardy
- 11,922
- 11
- 81
- 119
45
votes
2 answers
The formal p-adic numbers
The real numbers can be defined in two ways (well, more than two, but let's stick to these for now): as the Cauchy completion of the metric space $\mathbb{Q}$ with its usual absolute value, or as the Dedekind completion of the ordered set…
Mike Shulman
- 65,064
45
votes
2 answers
Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
npbool
- 563
45
votes
1 answer
Determinant of a determinant
Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m \times m$ determinant formula to get a single block,…
Bjorn Poonen
- 23,617
45
votes
39 answers
What out-of-print books would you like to see re-printed?
It's excellent news that the LMS are to re-publish Cassels & Fröhlich. There are many other excellent mathematics books which are just about impossible (or at least very expensive) to get hold of, though this problem seems to be getting a bit…
Martin Bright
- 4,155
45
votes
1 answer
Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?
Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation
$$
\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.
$$
Basic theorems about Dirichlet series imply that if the Dirichlet series on the right…
Jeremy Rouse
- 19,981
- 2
- 76
- 103
45
votes
0 answers
Extending a line-arrangement so that the bounded components of its complement are triangles
Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that $\{L_1,\dots,L_m\}$ is triangulating away from infinity iff every…
James Propp
- 19,363
45
votes
15 answers
What are the applications of hypergraphs?
Hypergraphs are like simple graphs, except that instead of having edges that only connect 2 vertices, their edges are sets of any number of vertices. This happens to mean that all graphs are just a subset of hypergraphs.
It strikes me as odd, then,…
DoubleJay
- 2,353
- 3
- 33
- 45