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Results tagged with open-problems
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user 9550
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 before, and with what results?"
25
votes
Series whose convergence is not known
Convergence of $\sum_{k=1}^\infty (-1)^k \frac{k}{p_k}$ is unknown, where $p_k$ is k-th prime (Guy 1994, p. 203; Erdős 1998; Finch 2003, according to Eric Weisstein's MathWorld).
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16
votes
Not especially famous, long-open problems which anyone can understand
Bonnessen—Fenchel conjecture: Which convex body of constant width has the least volume? Is it Meissner's tetrahedron?
11
votes
Not especially famous, long-open problems which anyone can understand
Let $R(x)=P(x)/Q(x)$ where $P(x)$ and $Q(x)$ are polynomials with integer coefficients and $Q(0)\neq 0$. Is there an algorithm that given $P(x)$ and $Q(x)$ as an input always halts and decides if the …
19
votes
Not especially famous, long-open problems which anyone can understand
Is there an upper bound of quotients in the continued fraction representation of $\sqrt[3]{2}=[ 1; 3, 1, 5, 1, 1, \dots]$?
10
votes
Not especially famous, long-open problems which anyone can understand
Given $n\in\mathbb N$, what is the smallest $k\in\mathbb N$ such that the harmonic number $H_k>n$?
It has been conjectured that for all $n$ the answer is $\lfloor\exp(n-\gamma)-1/2\rfloor$. See A00238 …
5
votes
Not especially famous, long-open problems which anyone can understand
What is the least $V$ such that any convex body of unit volume can be fit into a tetrahedron of volume $V$? It is known that $V \ge 9/2$ and conjectured that $V = 9/2$.
6
votes
Not especially famous, long-open problems which anyone can understand
What is the least $S$ (if any) such that any subset of a plane of area $S$ contains $3$ vertices of a triangle of unit area?
10
votes
Not especially famous, long-open problems which anyone can understand
Is there a positive integer which is both triangular and factorial except these obvious examples: $1, 6, 120$? (Tomaszewski conjecture, http://oeis.org/A000217)
10
votes
Not especially famous, long-open problems which anyone can understand
Is there a rectangle that can be cut into $3$ congruent connected non-rectangular parts?
4
votes
Not especially famous, long-open problems which anyone can understand
Are there infinitely many partition numbers divisible by $3$? See A000041.
7
votes
Not especially famous, long-open problems which anyone can understand
Is there such $n\in\mathbb{N}$ that ${^n\pi}\in\mathbb{N}$? (see tetration)
6
votes
Not especially famous, long-open problems which anyone can understand
Is Hilbert's tenth problem for Diophantine equations in rational numbers decidable?
Is Hilbert's tenth problem for Diophantine equations of power $3$ decidable?
Is there a universal Diophantine equat …
10
votes
Not especially famous, long-open problems which anyone can understand
What is the largest possible volume of the convex hull of a space curve having unit length?
40
votes
Not especially famous, long-open problems which anyone can understand
Is there a dense subset of a plane having only rational distances between its points?
8
votes
1
answer
329
views
Algorithmic decidability of equality in the ring of periods
Suppose two elements of the ring of periods are given by their systems of polynomial inequalities with rational coefficients. Is there a known algorithm deciding their equality? Is it known if their e …
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