Most active questions
181 questions from the last 7 days
12
votes
6
answers
1k
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Sum and sum of cubes are both perfect cubes
I am interested in integer tuples $(a_1, a_2, \dots, a_n)$ with $n > 3$, satisfying the following conditions:
Each $a_i \in \mathbb{Z} \setminus \{0\}$;
The $a_i$ are pairwise distinct and no two ...
- 133
18
votes
2
answers
1k
views
What is this modified arithmetico-geometric mean function?
I acquired a vintage programmable calculator and thought I'd give it a spin by computing some interesting transcendental function. I wanted to compute the arithmetico-geometric mean but the calculator ...
- 8,603
14
votes
3
answers
2k
views
Where did this theorem appear?
In a 1934 paper of Erdős and Turán , whose title is On a problem in the elementary theory of numbers, they said,
… Their proof depends on a theorem of Mr. Pólya asserting that if we denote by $q_1<...
- 697
5
votes
3
answers
615
views
Why isn’t the weak-* topology normed?
I know this is a basic question, but I have been struggling to find an answer!
The weak-* topology (say on $M(X)$, where $X$ is a compact separable metric space) is induced by the metric $$d(\mu,\nu)=\...
- 560
14
votes
1
answer
688
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An identity involving binomial coefficients
How to prove that
$$\sum _{j=0}^{n-i} \frac{(-1)^j \binom{n-i}{j}}
{(i+j) (i+j+1) (n+i+j+1)\binom{n+i+j}{n-i}}
=\frac{4 (2 i-1)!\, (2 n-2 i+1)!}{(2n+2)!},$$
where $n$ and $i$ are integers such $1\le i\...
- 135k
13
votes
1
answer
594
views
Where can I find a reference table for ordinal arithmetic?
I am working on a calculator for ordinal arithmetic.
Ordinal arithmetic is extremely tricky. So much so that there have been academic papers on the very subject that got some results wrong. Not to ...
- 231
1
vote
2
answers
283
views
The strong Mersenne conjecture
The strong twin conjecture:
For every number $a≥0$, there exist two prime numbers $p$ and $p+2$ such that $$a+4<p<2^{a+4}$$
and it was believed that this conjecture implies the twin conjecture.
...
- 1,215
6
votes
1
answer
473
views
Construction of algebra objects
We know that if $\mathcal{C}$ is an $\infty$-category, then $\operatorname{Fun}(\mathcal{C},\mathcal{C})$ is a monoidal $\infty$-category, whose tensor product is given by composition. It is well-...
- 509
6
votes
2
answers
352
views
Does the converse of Schur's lemma hold for smooth representations of finite length of p-adic group?
$\DeclareMathOperator{\Hom}{Hom}$Let $G$ be a p-adic group. Schur's lemma for smooth representations of $G$ states that if $\pi$ is an irreducible smooth representation of $G$, then $\dim \Hom_G(\pi,\...
- 161
4
votes
2
answers
354
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Finite pushforward of symmetric differentials
Let us consider a finite surjective morphism $f\colon X\to Y$ between smooth projective varieties. As we know, for each $r\leq\dim(X)=\dim(Y)$, there is an injection $f^*\colon \Omega_Y^{r}\to f_*\...
- 41
5
votes
1
answer
486
views
Closed form for A131823
Let
$T(n,k)$ be A131823, i.e., integer coefficients whose ordinary generating function for the $n$-th row is $ \displaystyle{ \prod_{i=0}^{n-1} (1 + x^{2^i})^{n-i} }$.
$\operatorname{wt}(n)$ be ...
- 5,890
7
votes
1
answer
283
views
The second Brauer-Thrall conjecture for finitely generated algebras
Let $k$ be an infinite field and $A$ a finitely generated $k$-algebra, so $A \cong k \langle x_1, \dots, x_n \rangle / I$. We say that $A$ is of
infinite type if there are infinitely many finite ...
- 730
6
votes
2
answers
266
views
Existence of collision-free assignment of points
Let $P = \{P_1, \ldots, P_N\}$ and $Q = \{Q_1, \ldots, Q_N\}$ be two sets of $N$ distinct points in $\mathbb{R}^n$. Each point in $P$ is to be assigned to a unique point in $Q$ and move toward it ...
- 61
2
votes
1
answer
382
views
Power series with integer coefficients: surprising facts, and one question
Let $\psi(x)$ be defined as follows, where $g$ is a stricly increasing real-valued function with $g(0)=0$.
$$
y = \psi(x) = \sum_{k=0}^\infty \lfloor g(k)\rfloor\cdot x^k
$$
When $x=\frac{1}{2}$, if $...
- 3,140
3
votes
1
answer
292
views
On the connection between chaos and ergodicity
This is a specific question pertaining to the 'universal' properties of chaos in dynamical systems.
Consider a continuous map $T:B\to B$, with $B\subset\mathbb{R}^n$ a compact subset. This defines a ...
- 1,131