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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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.
me too's user avatar
  • 8,030
3 votes
1 answer
128 views

(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 ...
Hugo van der Sanden's user avatar
1 vote
1 answer
45 views

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 ...
Kamal Saleh's user avatar
  • 6,403
1 vote
0 answers
301 views

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 ...
pie's user avatar
  • 3,297
0 votes
1 answer
34 views

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 ...
Dan's user avatar
  • 21.2k
-2 votes
0 answers
81 views

An interesting conjecture

This question is a very long one, so sit tight. I designed and wrote a bot (in c++) that plays blackjack. Without going into any unnecesary details, this is how it takes decisions: It takes 3 ...
Yağız Alp Ersoy's user avatar
0 votes
1 answer
94 views

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 &&...
Huatao Xue's user avatar
3 votes
2 answers
131 views

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 $\...
Daniel Donnelly's user avatar
13 votes
2 answers
269 views

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 ...
Dan's user avatar
  • 21.2k
17 votes
3 answers
850 views

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 &...
Dan's user avatar
  • 21.2k
3 votes
0 answers
66 views

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 ...
Dan's user avatar
  • 21.2k
2 votes
1 answer
108 views

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: ...
Miracle Invoker's user avatar
1 vote
0 answers
90 views

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 ...
Léo Mousseau's user avatar
0 votes
0 answers
113 views

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}^\...
Dylan Levine's user avatar
  • 1,575
0 votes
2 answers
82 views

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 ...
Mako's user avatar
  • 526
1 vote
1 answer
136 views

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-...
mick's user avatar
  • 15.9k
20 votes
1 answer
519 views

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 ...
Dan's user avatar
  • 21.2k
6 votes
1 answer
187 views

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 ...
pie's user avatar
  • 3,297
-3 votes
2 answers
189 views

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 ...
RBen's user avatar
  • 7
1 vote
0 answers
45 views

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. ...
Carlyle's user avatar
  • 2,817
12 votes
2 answers
238 views

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 = ...
Dylan Levine's user avatar
  • 1,575
0 votes
0 answers
27 views

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 ...
user avatar
23 votes
3 answers
1k views

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 \...
pie's user avatar
  • 3,297
2 votes
1 answer
145 views

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+...
pie's user avatar
  • 3,297
0 votes
0 answers
36 views

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)...
mick's user avatar
  • 15.9k
5 votes
1 answer
97 views

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},\...
Dan's user avatar
  • 21.2k
10 votes
2 answers
345 views

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 ...
Moaoly's user avatar
  • 153
6 votes
1 answer
126 views

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 $...
Dan's user avatar
  • 21.2k
20 votes
1 answer
398 views

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 ...
Dan's user avatar
  • 21.2k
0 votes
0 answers
134 views

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, ...
Nikita Dezhic's user avatar
0 votes
0 answers
40 views

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. ...
Blanco's user avatar
  • 656
0 votes
0 answers
63 views

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 ...
mick's user avatar
  • 15.9k
0 votes
1 answer
80 views

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 ...
SRobertJames's user avatar
  • 4,082
13 votes
0 answers
397 views

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 ...
Dan's user avatar
  • 21.2k
27 votes
4 answers
640 views

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 ...
Dan's user avatar
  • 21.2k
2 votes
0 answers
31 views

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) ...
SRobertJames's user avatar
  • 4,082
0 votes
0 answers
102 views

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 ...
mick's user avatar
  • 15.9k
7 votes
2 answers
202 views

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 ...
Dan's user avatar
  • 21.2k
22 votes
6 answers
1k views

$\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\beta} d\alpha d\beta=(0.9999999913...)(\pi/2)$? 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 ...
Dan's user avatar
  • 21.2k
1 vote
0 answers
37 views

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 ...
AndroidBeginner's user avatar
23 votes
1 answer
729 views

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 ...
Dan's user avatar
  • 21.2k
14 votes
2 answers
610 views

"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 ...
Dan's user avatar
  • 21.2k
3 votes
0 answers
102 views

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 ...
user avatar
1 vote
1 answer
156 views

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 ...
Dan's user avatar
  • 21.2k
34 votes
6 answers
2k views

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 ...
Dan's user avatar
  • 21.2k
0 votes
0 answers
28 views

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 ...
wurd's user avatar
  • 1
1 vote
1 answer
63 views

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 ...
mick's user avatar
  • 15.9k
26 votes
2 answers
1k views

A curious coincidence: $\prod\limits_{k=1}^\infty\left(1+\int_{k}^{k+1}\left(\frac{\sin (\pi x)}{x}\right)^2\mathrm dx\right)\overset{?}{=}\pi/2$

It is known that $\int_0^\infty \frac{\sin x}{x} \mathrm dx=\dfrac{\pi}{2}$ (proof) and $\int_0^\infty\left(\frac{\sin x}{x}\right)^2\mathrm dx=\dfrac{\pi}{2}$ (proof). Numerical investigation ...
Dan's user avatar
  • 21.2k
2 votes
2 answers
142 views

Does it make sense to divide mathematical theorems into those susceptible to experimental conjecture and those that are not?

I read the following in a good book: ”let's start from zero", the authors are Vinicio Villani and Maurizio Berni (pisan mathematicians) but I don't know if the book is also marketed outside Italy....
Fausto Vezzaro's user avatar
1 vote
0 answers
116 views

Ec primes multiple of 67 and $1\pmod{67}$

Ec primes are primes of the form: $(2^n-1)\cdot 10^d+2^{n-1}-1=ec(n)$, where d Is the Number of decimal digits of $2^{n-1}$. I conjecture that there are infinitely many primes of this form with n ...
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