Linked Questions

18
votes
63answers
11k views

What's your favorite equation, formula, identity or inequality? [closed]

Certain formulas I really enjoy looking at like the Euler-Maclaurin formula or the Leibniz integral rule. What's your favorite equation, formula, identity or inequality?
34
votes
24answers
6k views

Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...
68
votes
5answers
5k views

Is there a dense subset of the real plane with all pairwise distances rational?

I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory ...
84
votes
3answers
8k views

Convergence of $\sum(n^3\sin^2n)^{-1}$

I saw a while ago in a book by Clifford Pickover, that whether $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open. I would think that the question of its convergence is really ...
31
votes
6answers
3k views

Status of PL topology

I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
52
votes
4answers
2k views

What is the limit of gcd(1! + 2! + … + (n-1)! , n!) ?

Let $s_n = \sum_{i=1}^{n-1} i!$ and let $g_n = \gcd (s_n, n!)$. Then it is easy to see that $g_n$ divides $g_{n+1}$. The first few values of $g_n$, starting at $n=2$ are $1, 3, 3, 3, 9, 9, 9, 9, 9, ...
17
votes
9answers
4k views

Open problems in Euclidean geometry ?

Which are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a ...
29
votes
3answers
4k views

Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1. It is easy to show that $$\sum_{1 \leq k } ...
20
votes
4answers
1k views

Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$ a generalized polynomial if for any distinct integers $m$ and $n$ we have $(m - n)|(f(m)-f(n))$. It is easy to check that polynomial ...
23
votes
3answers
657 views

Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
12
votes
2answers
1k views

Induction, the infinitude of the primes, and workaday number theory

There are various open problems in the subject of logical number theory concerning the possibility of proving this or that well-known standard results over this or that weak theory of arithmetic, ...
7
votes
9answers
2k views

Not especially famous, long-open problems which higher mathematics beginners can understand

This is a pair to Not especially famous, long-open problems which anyone can understand So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
3
votes
2answers
447 views

Emptyness of a projective variety

Let $S$ be some (fixed) subset of $\mathbb{Z} [X_1, \dots , X_n]$ which contains only homogeneous polynomials, and if $F$ is a field, let $X(F)$ be the set of $ x \in P^{n-1}(F)$ such that $f (x) = 0$ ...
15
votes
1answer
1k views

On a Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number ...
15
votes
0answers
2k views

Open problems/questions in representation theory and around ?

What are open problems in representation theory ? What are the sources (books/papers/sites) discussing this ? Any kinds of problems/questions are welcome - big/small, vague/concrete. Some estimation ...

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