Questions tagged [conjectures]
For questions related to conjectures which are suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found
1,385 questions
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Conjecture: If a tangential quadrilateral's incenter is its lamina centroid, then it must be a kite.
For symmetry reasons it is clear that rhombi do have the "incenter = lamina centroid" property. It is also not difficult to find some special kites having this property. And here the open ...
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Is there a simple proof of this metric relation in this Sangaku?
Here's a sangaku problem I've been imagining but I don't know if it's already known :
(ABCD) is a parallelogram whose diagonals intersection at O .
If : a , b , c and d are the radii of the circles ...
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Does $\sum_{n=0}^{\infty}\frac{(4n)!^2}{2^{8n}(n+1)^2(2n)!^4}$ have the proposed closed form?
Context
While working with the $_4F_3(a,b,c,d;e,f,g;x)$ I arrived to:
$$S=\frac{64}{9}\left(\hspace{.1cm} _4F_3(-\frac{3}{4},-\frac{3}{4},-\frac{1}{4},-\frac{1}{4};-\frac{1}{2},-\frac{1}{2},1;1)-1\...
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Given an $n$-gon, by how much must we increase the length of each of $k$ chosen sides so that they form a $k$-gon?
The polygon inequality states that the sum of any $n-1$ sides of a $n$-gon greater than the $n$-th side.
Let $n \ge 4$ and $3 \le k < n$. Let a $n$-gon have positive side lengths $
a_1 \le a_2 \le \...
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Conjecture: a constant-free equation is solvable in a group $G$ if and only if it is solvable in its generating set $B$ for any $G=\langle B\rangle$
Conjecture.
Let $G$ be a group and $B$ any set of generators for $G$. That is to say $G = (G, \cdot) = \langle B \rangle$. Then for any equation $E=F$ in $G$ that is constant-free, we have that $E=...
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Is the area of a cyclic polygon of nine or more sides greater than the square of the geometric mean of the sides?
The side lengths of a convex polygon do not uniquely determine the shape of the polygon but if the polygon is cyclic then the shape is uniquely determined by the side lengths. Consider a cyclic $n$-...
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If $H$, $K$ and $L$ are subgroup such that $H$ commutes with $K$ and $L$ then is $H\ast K\ast L$ a subgroup?
If $H$ and $K$ are subgroup of a group $(G,\ast,e)$ then I know that $H\ast K$ is a subgroup of when $H$ is commutable with $K$: so I am searching a counterexample showing that if a subgroup $X$ ...
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Does there exist a number such that the first 16 numbers of its Collatz sequence mod 4 contain four 1s, 2s, 3s, and 4s?
I was setting a sudoku with a very unique constraint and I came across this. I thought it wouldn't be that hard to find a number, but after writing a python program to find one, I am doubtful of its ...
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A “block” generalization of factorions
I’ve been exploring a generalization of factorions—that is, numbers equal to the sum of factorials of their digits—and wanted some feedback on the concept.
Let $n \ge 1$ where $n$ is an integer. A ...
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Rational points and sections on a family of genus-3 hyperelliptic curves
I am interested in the proof or disproof of some conjectures about rational points and sections over $\mathbb{Q}$ for the following family of genus-3 hyperelliptic curves:
$$
C_t: f(x,a)\, g(x,a)\, h(...
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$a_1,\dots,a_n$ periodic sequences summing to $p_1,\dots,p_n$ over each of their resp. periods, then their sums synch. to some value $\leq\sum_i p_i$.
Conjecture.
Let $(a_i(j))_{j \geq 0}$ be sequences of natural numbers $\geq 1$. For example $a_1 = \overline{2} = 2,2,2,2, \dots$, is the constant $2$, but $a_3 = \overline{2,1,2}$ is not.
Define $B =...
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A conjecture about a difference expression of $n$ positive numbers
The starting point for this question is a set of $n$ positive ordered numbers:
$$ x_{1} \, \ge \, x_{2} \, \ge \, \ldots \, \ge x_{n} \, > \, 0 \; .$$
From these numbers difference expressions ...
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Does decomposition of PDFs guarantee independence of random variables?
Is this conjecture correct? If not, can it be modified to a correct one:
Let $X,Y$ be continuous RVs with joint PDF $f(x,y)$. Then $X,Y$ are independent iff there exists functions $g, h$ such that $$...
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A similar sequence to OEIS-A098021
Let $U_0=0$, we make $U_n$ by the following rules :
If $\color{magenta}n=U_k$ for some $k \in \mathbb{N}$ ,
$$U_{n+1} = U_{\color{magenta}n} + \color{red}5$$
$$U_{n+2} = U_{n+1} + \color{red}5$$
...
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Do the area, semiperimeter and longest side of a cyclic quadrilateral always satisfy the triangle inequality?
My experimental data supports the following conjecture. Can this be proved?
Conjecture: Let a convex quadrilateral be inscribed in a circle of radius $R>0$. Let $a$ denote the longest side, $s$ ...
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