If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it ...
7
votes
5answers
1k 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 ...
243
votes
4answers
18k views
Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?
Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\ $ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
104
votes
75answers
13k views
Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...
151
votes
8answers
9k views
Polynomial representing all nonnegative integers
Lagrange proved that every nonnegative integer is a sum of 4 squares.
Gauss proved that every nonnegative integer is a sum of 3 triangular numbers.
Is there a 2-variable polynomial $f(x,y) \in ...
22
votes
3answers
2k 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 } ...
32
votes
3answers
5k views
Do there exist chess positions that require exponentially many moves to reach?
By "chess" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an ...
13
votes
0answers
1k 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 ...
25
votes
3answers
995 views
Are most cubic plane curves over the rationals elliptic?
%This is a new version of the original question modified in the light of the answers and comments.
The word 'most' in the title is ambiguous. The following is one way of making it precise.
...
20
votes
4answers
6k views
Does pi contain 1000 consecutive zeroes (in base 10)?
The motivation for this question comes from the novel Contact by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at ...
8
votes
0answers
206 views
Are there only finitely many maximal subfactors of a fixed finite index ?
A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
Question: are there only finitely many maximal subfactors of a fixed finite ...
14
votes
2answers
606 views
Minimal graphs with a prescribed number of spanning trees
As its long ago since Erdős died and mathoverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem ...
2
votes
2answers
256 views
If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?
Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real"(symmetric) ? And vice verse ?
...
15
votes
1answer
409 views
Convex bodies with constant maximal section function in odd dimensions
In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ ...
9
votes
1answer
315 views
Integer-distance sets
Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...
7
votes
0answers
199 views
Why would dim primitive irrep divide size of some conjugacy class ?
From Isaacs et.al. 2005
Conjecture C. Let χ be a primitive
irreducible character of an arbitrary
finite group G. Then χ(1) divides |
clG(g)| for some element g ∈ G.
Here, of course, we ...
