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
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An extension of Brahmagupta's theorem.
As we know, there are conjectures that are easy to formulate but difficult to prove, and there are conjectures that are easy to prove but difficult to conceive. This conjecture is simple to conceive ...
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Conjecture about Proving Primality of Fermat numbers by Elliptic Curves technic
In 2008 and 2009, Denomme-Savin and Tsumura provided 2 papers providing a Primality Test for Fermat numbers based on Elliptic Curves technic:
$$ \text{Let } DST(x)= \frac{\displaystyle x^4+2x^2+1}{\...
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Are there infinitely many primes that are a highly composite number $\pm 1$?
I've looked at some highly composite numbers and realized that a lot of them are almost primes, i.e. differ only by $1$ to the next closest prime. Here's a short list (I made) of highly composite ...
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For any square-free $n \geq 1$ and $a \in \Bbb{Z}/n$ including $\gcd(a,n) \neq 1$, then in the list $a, a^2, a^3, \dots$ either $a$ or $a^2$ repeats?
For example, modulo $30$ we have that $\gcd(5, 30) = 5$, but $5, 5^2, 5^3 = 5, 5^2, \dots$ goes the list, so both repeat.
We know it's true when $\gcd(a,n) = 1$ because a cyclic group is formed in $\...
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Is there a perfect group in which not every element is a commutator?
Is there a perfect group in which not every element is a commutator?
By a well-known fact, it must have order at least $96.$
By Ore's conjecture (now a theorem), it must be infinite or non-simple.
3
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(dis)proof of conjecture on square unit fractions
Consider a finite set S of positive integers, and define $q(S) = \sum_{s \in S}{1/s^2}$. Letting $\rho = \pi^2/6$, we have $q(S)$ in the ranges $[0, \rho - 1), [1, \rho)$. I conjecture that for every ...
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On the fractional parts of the roots of the Alternating Harmonic Numbers
We define $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-\ln2\right)$$As the $x$th Alternating Harmonic Number (test out a few values to see why). Let $x_n$ be the $n$th ...
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A conjecture on representing $\sum\limits_{k=0} ^m (-1)^ka^{m-k}b^k$ as sum of powers of $(a+b)$.
UPATE: I asked this question on MO here.
I was solving problem 1.2.52 in "An introduction to the theory of numbers by by Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery"
Show that if ...
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True or false: In a 2D Poisson process, for every point $P$, there exists a convex $1000$-gon with Poisson points as vertices, that contains only $P$.
I made a Desmos graph that generates $30$ uniformly random black points in a disk, with the centre of the disk in red.
I asked myself, "Can I always draw a convex quadrilateral with four of the ...
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Conjecture: For any prime $n$, at least one of $2n^2-1$ , $2n^2+1$, or $\sqrt{2n^2-1}$ is prime.
In my spare time, I came up with this conjecture:
For any prime $n$, at least one of $2n^2-1$ , $2n^2+1$, or $\sqrt{2n^2-1}$ is prime.
Examples:
\begin{alignat}{2}
2 &\to\quad 2\cdot 2^2-1 &&...
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New conjecture? $(\varphi(n))! = -1 \pmod n \iff n$ is prime (nearly the same as Wilson's) [duplicate]
$$
(n-1)! = -1 \pmod n \text{ iff } n \text{ is prime}, \text{ is Wilson's theorem,}
$$
But coincidentally for now the expression passed to factorial is $n - 1$ which is (iff $n$ is prime) equal to $\...
14
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Conjecture: Expected total area of a certain set of random triangles in a unit disk is $1/\pi$.
Choose $3n$ independent uniformly random points in a disk with perimeter $x^2+y^2=1$. Label the points $P_1,P_2,P_3,\dots,P_{3n}$ in order of increasing $x$-coordinates. Form triangles $\triangle ...
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Three random points on $x^2+y^2=1$ are the vertices of a triangle. Is the probability that $(0,0)$ is inside the triangle's incircle exactly $0.13$?
Three uniformly random points on the circle $x^2+y^2=1$ are the vertices of a triangle.
What is the probability that $(0,0)$ is inside the triangle's incircle?
(This a variation of the question &...
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Conjecture about average cluster size in a prime-like sequence
Consider the sequence $u_n=n^a+p_n$, where $a$ is a real constant and $p_n$ is the $n$th prime.
Write down the terms in $u_n$ increasing from left to right. From each term, draw a line segment ...
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Moments of Alternate Elliptic Integrals
Let $K_s$ denote Ramanujan's Alternate Elliptic Integrals as follows:
$$K_s:=K_s(k)=\frac{\cos(\pi s)}{2}\int_0^1\frac{t^{s-1/2}}{(1-t)^{1/2+s}(1-k^2t)^{1/2-s}}dt$$
Where in Hypergeometric Form it is:
...
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Could the smooth 4 dimensional poincare conjecture be independent of ZFC?
The Smooth 4-dimensional Poincare conjecture ($S4PC$) states that any closed smooth 4-dimensional manifold which is homeomorphic to $S^4$ will be diffeomorphic to $S^4$.
My question is wether this ...
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Average value of random infinite series
Let $u(b)$ be a uniformly random number bounded by $0\le u(b)\le b$ and let $v(n)=\underbrace{u\circ u\circ...\circ u}_{n\text{ times}}\phantom{ }\circ u(1)$. What on average is $\sum\limits_{n=0}^\...
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Values of $\Phi_n(-1)$
Let
$$\Phi_n(x) = \prod_{0<k\leq n, \gcd(k,n)=1}(x-e^{\frac{2\pi i k}{n}})$$
be the $n$-th cyclotomic polynomial. By observation it seems that cyclotomic polynomials when evaluated at $x=-1$ give ...
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Primes of the form $q_n = p^n - 2$?
Does this conjecture have a name ?
I assume it is true but has not been proven or disproven.
I am aware of prime twins, Dickson's conjecture, Bunyakovsky conjecture, Schinzel's hypothesis H, Bateman-...
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Conjecture: If $A,B,C$ are random points on a sphere, then $E\left(\frac{\text{Area}_{\triangle ABC}}{\text{Area}_{\bigcirc ABC}}\right)=\frac14$.
On (not in) a sphere, choose three independent uniformly random points $A,B,C$. Is the following conjecture true:
The expectation of the ratio of the area of (planar) $\triangle ABC$ to the area of ...
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Conjecture: $\binom{n}{k } \mod m =0$ for all $k=1,2,3,\dots,n-1$ only when $m $ is a prime number and $n$ is a power of $m$
While playing with Pascal's triangle, I observed that $\binom{4}{k } \mod 2 =0$ for $k=1,2,3$,and $\binom{8}{k } \mod 2 =0$ for $k=1,2,3,4,5,6,7$ This made me curious about the values of $n>1$ and ...
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Could a sequence in the Collatz conjecture actually increase without bound?
If my understanding is correct, than the Collatz conjecture could only be false if there is at least two closed cycle in it or if there is a number which increases without bound.
$3x-1$
We know that ...
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Conjecture about boundary commuting with union
Given a Topological space $(X,\tau)$ and two disjoint open sets $A,B$, the following holds:
$$\partial(A\cup B) = \partial A\cup \partial B$$
I am interested in a modified version of this.
...
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Does $\int_{1}^{2}\ln\left(x+\ln\left(x^2+\ln\left(x^3+\ln\left(\dots\right)\right)\right)\right)dx=1$?
I was messing around with the infinitely nested logarithm $f(x)=\ln\left(x+\ln\left(x^2+\ln\left(x^3+\ln\left(\dots\right)\right)\right)\right)$ on Desmos when I decided to take the integral from $x = ...
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For every abelian variety $A$ over a number field $K$ and for a prime number
For every abelian variety $A$ over a number field $K$ and for a prime number $p$ ,$$\begin{equation*} (-1)^{\operatorname{rk}_p (A/K)} = w_{A/K}. \end{equation*}$$
2-parity conjecture holds for all ...
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Conjecture: $\sum\limits_{k=1}^nk^m=S_3(n)\times\frac{P_{m-3}(n)}{N_m}$ for odd $m>1 \ ;\ =S_2(n)\times\frac{P_{m-2}'(n)}{N_m}$ for even $m$.
When I was in high school, I was fascinated by $\displaystyle\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}$ so I tried to find the general solution for $\displaystyle\sum\limits_{k=1}^n k^m$ s.t $m \in \...
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A generalization of the Stolz-Cesàro: For $k\ge 1$, $\lim_{n\to\infty }\frac{a_{n+k}-a_n}{b_{n+k}-b_n}=l$ implies $\lim_{n\to\infty}\frac{a_n}{b_n}=l$
Stolz-Cesàro theorem case $\frac{*}{\infty}$:-
If $b_n $ is a monotone increasing sequence and $\lim \limits_{n \to \infty} b_n = \infty $,
and if $\lim \limits_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+...
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A weaker version of the second Hardy-Littlewood conjecture.
Let $n$ be a positive integer and let $f(n)$ be the counting function for non-composite numbers.
So
$$f(0)=0,f(1)=1,f(2)=2,f(3)=3,f(4)=3,f(5)=4$$
Now my mentor noticed as a kid that apparantly
$$f(x+y)...
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Does this recursive journey through Pascal's triangle always reach $1$?
Let $p(k)$ be the $k^\text{th}$ number in Pascal's triangle, numbered from left to right in each row, going down the rows.
So, for example, $p(1)$ to $p(10)$ are $\binom{0}{0},\space \binom{1}{0},\...
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Conjecture: If $\sum_{i=1}^n \frac{1}{x_i}=1$ then $x_i | x_j$
Let $x_1,x_2,x_3,\cdots , x_n\in\mathbb{N}$ prove that if $\sum_{i=1}^n\frac{1}{x_i}=1$, then there exist $x_j,x_i$ such that $x_j | x_i$.
I read that I should avoid the no clue questions, but this is ...
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Conjecture: $\prod\limits_{k=0}^{n}\binom{2n}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2k}{k}$ only if $n=1,2,5$.
The diagram shows Pascal's triangle down to row $10$.
I noticed that the product of the blue numbers is divisible by the product of the orange numbers. That is (including the bottom centre number $...
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Take a random walk on Pascal's triangle, without revisits: Does the final number have infinite expectation?
Let's take a random walk on Pascal's triangle, starting at the top. Each number is in a regular hexagon. At each step, we can move to any adjacent hexagon with equal probability, but we cannot revisit ...
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A question about a paper by Gilmer on union-closed sets conjecture
In the recent paper by Justin Gilmer, the following identity is assumed to hold: given finite random variables $X, X' \in \{0,1\}; C, C' \in \{1,..,M\}$, such that the pairs $(X, X')$, $(C, C')$, $(X, ...
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Are there some famous conjectures related to perfect graphs?
I am reading the book "Graph Classes: A Survey", and I know that Perfect Graph Conjecture (PGC) and Strong Perfect Graph Conjecture (SPGC) once were famous conjectures about perfect graphs.
...
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An infinite linear system of equations with an uncountable number $A$ of equations
I will start with an example to make things clear and avoid confusion :
Take all $x>0$ and
$$\exp(x) = \sum_{-1<n} a_n x^n$$
Now finding $a_n$ can be described as an infinite linear system of ...
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When does the magnitude of the gradient equal the surface area of the $dxdy$ patch?
Given a surface $S$ in $\mathbb R^3$, what is the relationship between the gradient (when $S$ is defined as a level curve of function $F: \mathbb R^3 \to \mathbb R$) and surface area?
I noticed such a ...
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Can this be done? Split Pascal's triangle (without the $1$s) with a straight line into two regions of equal sums.
Consider Pascal's triangle with $n$ rows, without the $1$s, with each number corresponding to a vertex on a pyramid of equilateral triangles, as shown below with example $n=5$.
Can the triangle be ...
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Conjecture: In Pascal's triangle with $n$ rows, the proportion of numbers less than the centre number approaches $e^{-1/\pi}$ as $n\to\infty$.
Consider Pascal's triangle with $30$ rows (the top $1$ is the $0$th row). The centre number is the number in the middle of row $30\times \frac23=20$, which is $\binom{20}{10}=184756$. The proportion ...
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What is the change in angle of of a symmetric matrix when summed over a plane?
A symmetric matrix has orthogonal eigenvectors with real eigenvalues, and hence can be thought of as scaling along a particular orthogonal set of axes.
Of course, not all vectors are (usually) ...
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0
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Are there modular arguments for Legendre's conjecture?
Legendre's conjecture says that there is always a prime between $n^2$ and $(n+1)^2$ for every positive integer.
See : https://en.wikipedia.org/wiki/Legendre%27s_conjecture
Now I wonder why people ...
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Conjecture: If $x_k$ are random in $(0,\pi/2)$ then expectation of $\frac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$ is $(\pi/2)^{2n-6}$ for $n>2$.
Let $E(n)=\text{expectation of }\dfrac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$ where $x_k$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.
Is the following ...
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$\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\beta} d\alpha d\beta=(0.9999999913...)(\pi/2)$? Seriously?
In the diagram, $\alpha$ and $\beta$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.
What is $\mathbb{E}(h)$?
Superimposing a cartesian coordinate system, the ...
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Large sets and Erdős-discrepancy
Large Sets
Erdos conjecture
I have a conjecture that is stronger than the Erdos discrepancy conjecture, can someone think of a counter example?
Let $S$ be any large set and let $(x_1,x_2,...)$ be any ...
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Does the interior of Pascal's triangle contain three consecutive integers?
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
A006987 (which is described here) is a list of the ...
14
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2
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"Twin binomial coefficient conjecture": There are no binomial coefficients that differ by $2$, excluding $\binom{n}{1}$ and $\binom{n}{n-1}$.
Consider the binomial coefficients, excluding those of the form $\binom{n}{1}$ and $\binom{n}{n-1}$.
There are some that differ by $1$, for example $\binom{7}{2}-\binom{6}{3}=21-20=1$.
There are some ...
3
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Poulet numbers of the form $3p$, where $p$ is a palindrome
A Poulet-number is a composite number $N$ satisfying $$2^{N-1}\equiv 1\mod N$$ A palindrome is a positive integer with a digit string in base $10$ which remains the same if it is written down ...
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Conjecture: Given any $2n+1$ points, we can always draw $n$ non-intersecting circles whose diameter endpoints are $2n$ of those points.
Is the following conjecture true or false:
Given any $2n+1$ coplanar points, we can always draw $n$ non-intersecting circles coplanar with the points, whose diameter endpoints are $2n$ of those ...
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6
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Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least one pair of non-intersecting circles coplanar with the points, such that two of the given points ...
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0
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Open problems in Lie symmetry theory
What are some famous open problems and conjectures in Lie symmetry theory? Is there some kind of list of these problems available or possibly a historical survey of the development of theory of ...
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If $n$ is a Poulet number then $n+2$ is not a Poulet number?
Consider Fermat pseudoprimes to base $2$, also called Sarrus numbers or Poulet numbers.
Inspired by prime twins it makes sense to consider :
Conjecture :
If $n$ is a Poulet number then $n+2$ is not a ...