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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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Conjecture on Infinitely Many Consecutive Pairs of Early Primes

An early prime is one which is less than the arithmetic mean of the prime before and the prime after. Conjecture: There are infinitely many consecutive pairs of early primes MY attempt Well, the fact ...
Saucitom's user avatar
5 votes
0 answers
116 views
+50

Conjectures involving $\Lambda(n)$

As the title suggests, I am looking for conjectures involving the Von Mangoldt function, $\Lambda(n)$. I understand this is not a rigorous mathematical question, however if reference requests for ...
Mako's user avatar
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1 vote
0 answers
79 views

Lower bound of $n$ th Taxi-cab number $N = a^3 + b^3 = x^3 + y^3$

Let $N,a,b,x,y$ be distinct positive integers such that $$N = a^3 + b^3 = x^3 + y^3$$ Also known as Taxicab numbers or Taxi-cab numbers. see also : https://oeis.org/A001235 let $t(n)$ be the $n$ th ...
mick's user avatar
  • 16.4k
1 vote
1 answer
109 views

Conjectures about the greatest common divisor of a vertical column of the pascal triangle.

I was playing around with pascal triangle I noticed an interesting property concerning the greatest common divisor $gcd$ of binomial coefficients along a vertical line. Specifically, the line ...
pie's user avatar
  • 6,456
10 votes
0 answers
370 views

Does the Sequence formed by Intersecting Angle Bisector in a Pentagon converges?

Now asked on MO here. Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $A_{n+1}B_{n+...
pie's user avatar
  • 6,456
3 votes
2 answers
146 views

A conjecture about the sum of Faulhaber polynomials mod 2.

Context: In this question I asked about "Finding a polynomial function for alternating $m$ numbers of odds and even numbers in sequences" I noticed that all Faulhaber polynomials are ...
pie's user avatar
  • 6,456
2 votes
1 answer
32 views

Finding measurable subsets of any given value?

I'm not sure if the following is true, but I would hope it is with the regularity properties of $\mu$. Let $X$ be a locally compact Hausdorff space with $\mu$ a nonzero Radon measure on $X$. Then ...
Isochron's user avatar
  • 1,399
2 votes
0 answers
52 views

Implications of having access to the Busy Beaver oracle

Apologies if I'm asking a naïve question as I've only recently learned about the concept. What would be the practical implications (if any) of having access to a magical black box providing the ...
mavzolej's user avatar
  • 1,472
1 vote
0 answers
62 views

Constructing a Curve from a Differentiable Function's Normal Lines at a Fixed Distance. [duplicate]

I've been experimenting with a graphing calculator and stumbled upon an interesting geometric construction involving functions and their normal lines. Here's the problem I'm trying to solve: Let's ...
pie's user avatar
  • 6,456
1 vote
1 answer
41 views

These constrained alternating series always satisfy an inequality.

Let $x_0 \in \Bbb{N}$ and suppose that $x_1 \lt \frac{x_0}{2}$, while $0 = x_n \leq \dots \leq x_3 \leq x_2\leq x_1$. Then is it possible that: $$ \sum_{i = 0}^n (-1)^i x_i \gt 1 $$ no matter what ...
SeekingAMathGeekGirlfriend's user avatar
-3 votes
1 answer
64 views

Weaker than $X^2 + 1$ conjecture: $(a + bX)^2 + 1$ is a prime number at least once for any $\gcd(a,b) = 1$. [closed]

Let $\gcd(a,b) = 1$ where $a, b\in \Bbb{Z}$ and consider $f(X) = X^2 + 1$ and the forward image of the coset $a + b\Bbb{Z}$ under $f$. There is always at least one prime number in $f(a + b\Bbb{Z})$. ...
SeekingAMathGeekGirlfriend's user avatar
-2 votes
1 answer
514 views

Rewriting in a special case that Brocard's problem have only finite primitive solution i.e Brown's numbers

I recently found a possible rewirting in the affirmative of the most famous Brocard problem or Ramanujan-Brocard problem: Problem : Let $n>3$ and $m>1$ be integers then $$(n(n+1)+1)!+1\neq m^2$$...
Ranger-of-trente-deux-glands's user avatar
0 votes
0 answers
36 views

is $\int_0^t \int_0^s \delta(u - v) \, du \, dv = \min(t, s) $? [duplicate]

Is the following integral equation true? $ \int_0^t \int_0^s \delta(u - v) \, du \, dv = \min(t, s) $ where $\delta$ is the Dirac delta function.
user1326164's user avatar
1 vote
0 answers
43 views

is there a function $\phi$ such that $\int_R \int_R f(t)g(s)\phi(t,s)dsdt=\int_Rf(t)g(t)\rho(t)dt $? [closed]

Let $\rho$ be a positive function, is there exist a function $\phi$ such that for any $f,g$: $$\int_R \int_R f(t)g(s)\phi(t,s)dsdt=\int_Rf(t)g(t)\rho(t)dt $$ of course both integrals are supposed ...
user1326164's user avatar
0 votes
1 answer
68 views

Application of Bertrand's postulate [duplicate]

We can use Bertrand's Conjecture ( that for any integer $n \not= 0$, there exists at least one prime number $𝑝$ with $n < p \leq 2n$ ) to demonstrate the ...
Alexis J's user avatar
-1 votes
1 answer
86 views

Is there a function $\phi$ such that $\int_0^t\int_0^t \phi(u,v) dudv=\int_0^t\psi(s)ds$? [closed]

Let $t$ be a positive real number and $\psi$ a positive function. I am looking for a function $\phi$ such that: $$\int_0^t \int_0^t \phi(u,v) du\,dv=\int_0^t\psi(s)ds$$ Does such a function exist or ...
user1326164's user avatar
6 votes
2 answers
396 views

Is the number 3 in the Collatz conjecture arbitrary?

One of the most famous conjectures in mathematics is the Collatz conjecture also known as $3n+1$ but my question is why we multiply the odd number by 3? I get that the conjecture probably wants to ...
pie's user avatar
  • 6,456
2 votes
1 answer
51 views

Graph isomorphism checking/detection for directed acyclic graphs

The graph isomorphism problem is hard for an arbitrary graph, and certain classes of graphs have been proven to be "GI-complete", which as I understand means they can be reformulated in ...
John Cataldo's user avatar
  • 2,649
2 votes
0 answers
183 views

Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions

Now asked on MO here. Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, $B_{n+1}$ is ...
pie's user avatar
  • 6,456
14 votes
2 answers
478 views

Conjectured connection between $e$ and $\pi$ in a semidisk.

A semidisk with diameter $\dfrac{e}{\pi}n$ is divided into $n$ regions of equal area by line segments from a diameter endpoint. Here is an example with $n=6$. Consider the $n$ arcs between ...
Dan's user avatar
  • 25.7k
0 votes
1 answer
75 views

A continuous nowhere differentiable function with $\alpha-$derivative exists at $\sup\{a\}$.

In This question I asked: Define $g(x)= |x|$ for $|x|\in [-1,1]$ , $g(x+2)=g(x)$ $$f(x)= \sum_{n \ge 1} \frac{3^n g\left(4^n x\right) }{4^n}$$ what is the $\sup \{\alpha\}$ such that $$ \lim\limits_{...
pie's user avatar
  • 6,456
3 votes
1 answer
104 views

For any positive integer $n\geq5$, there is at least one integer $m>n$ such that $ \pi\left(nm\right)=n+m $

Conjecture: For any positive integer $n\geq5$, there is at least one integer $m>n$ such that $ \pi\left(nm\right)=n+m $, where $\pi\left(x\right)$ is the prime counting function. Examples: $$ \pi(5\...
François Huppé's user avatar
9 votes
1 answer
271 views

A probability involving side lengths of a random triangle on a disk: Is it really $\frac37$?

Choose three uniformly random points on a disk, and let them be the vertices of a triangle. Call the side lengths, in random order, $a,b,c$. What is $P(a^2<bc)$ ? A simulation with $10^7$ such ...
Dan's user avatar
  • 25.7k
16 votes
1 answer
636 views

A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?

The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$. The petals are randomly chosen, but they must be ...
Dan's user avatar
  • 25.7k
2 votes
0 answers
159 views

What is $\sup\{a\}$ such that for continuous non differentiable function $f$ $\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h^a}$ exist?

Now asked on MO here The definition of the derivative of a function $f$ at a point $x$ is : $\lim\limits_{h \to 0 }\frac{f(x+h) -f(x)}{h}$ but what if we change that $h$ to be any continuous ...
pie's user avatar
  • 6,456
12 votes
1 answer
230 views

The vertices of a pentagram are five random points on a circle. Conjecture: The probability that the pentagram contains the circle's centre is $3/8$.

The vertices of a pentagram are five uniformly random points on a circle. Is the following conjecture true: The probability that the pentagram contains the circle's centre is $\frac38$. (The ...
Dan's user avatar
  • 25.7k
15 votes
2 answers
523 views

The vertices of a hexagon are random points on a unit circle; $a,b,c$ are the lengths of three random sides. Conjecture: $P(ab<c)=\frac35$.

The vertices of a hexagon are uniformly random points on a unit circle; $a,b,c$ are the lengths of three distinct random sides. A simulation with $10^7$ such random hexagons yielded a proportion of $0....
Dan's user avatar
  • 25.7k
8 votes
2 answers
245 views

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

Now asked on MO here. Given the length of the sides of a quadrilateral $a,b,c,d$ the area of the quadrilateral is less than or equal to $\frac{(a+b+c+d)^2}{16}$ i.e it is an upper bound of the area ...
pie's user avatar
  • 6,456
0 votes
1 answer
48 views

How do I just this conjecture?

I have 6 equations: $X_1=As+B_1t_1+B_2t_2$ $X_2=As+B_1t_2+B_2t_1$ $X_3=A_3s+B_3t_3$ $Y_1=C(X_1+t_1)$ $Y_2=C(X_2+t_2)$ $Y_3=D(X_3+t_3)$ where $X_1,X_2,X_3,Y_1,Y_2,Y_3,s,t_1,t_2,t_3$ are variables and ...
anonymous 's user avatar
2 votes
0 answers
132 views

Consecutive numbers

Are there ever more consecutive composite numbers than there are primes up to that point? I imagine not, because the primes are the ones which cancel out multiples, so will inevitably have to fill in ...
Talon Eaglefeathers's user avatar
5 votes
1 answer
137 views

Conjecture: Two different random triangles (both based on random points on a circle) have the same distribution of side length ratios.

On a circle, choose three uniformly random points $A,B,C$. Triangle $T_1$ has vertices $A,B,C$. The side lengths of $T_1$ are, in random order, $a,b,c$. Triangle $T_2$ is formed by drawing tangents to ...
Dan's user avatar
  • 25.7k
5 votes
0 answers
131 views

How do you find constants in an k-Tuple conjectures?

By introducing modular objects associated to the sequel of rings $$(Z/2,Z/6,Z/30,Z/210,Z/2310,..., Z/p_n\#Z)$$ a sequence of coefficients is updated$$(2;\color{green}{\frac83}; 3.2;...$$ (see my ...
Stéphane Jaouen's user avatar
4 votes
2 answers
237 views

Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $

Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I ...
pie's user avatar
  • 6,456
2 votes
0 answers
74 views

If $\lim\limits_{x\to\infty}\frac{f(x+1)-f(x)}{g(x+1)-g(x)}=l$ what are sufficient conditions to make $\lim\limits_{x\to\infty}\frac{f(x)}{g(x)}=l$?

In this question it is proved in the answers that If $f :[0 , \infty ) \to \mathbb{R}$ and $f $ is bounded on every $(a,b)$ such that $a<b <\infty$, prove that $\lim\limits_{x \to \infty }(f(x+...
pie's user avatar
  • 6,456
0 votes
0 answers
80 views

The Consecutive Composite Conjecture (Proof required)

I have been working on a conjecture the past couple days and would like assistance in determining whether or not it is true or not. The question is as follows. Given a distinct set of ascending prime ...
Michael Franklin's user avatar
2 votes
0 answers
60 views

Artin's conjecture on primitive roots for perfect powers

Let $a\neq -1,0,1$ be an integer. Write $a=(b^2c)^k$, where $b^2c$ is not a perfect power, and $c$ is squarefree. Artin's conjecture on primitive roots states that the asymptotic density of the set of ...
Jianing Song's user avatar
  • 1,923
1 vote
1 answer
41 views

Is there any material on this simpler version of Littlewood's Conjecture?

Littlewood's conjecture is defined as follows: For any real number $x$, define $f(x)$ as the distance between $x$ and the integer nearest to $x$, or $f(x) = \min(x - \lfloor x \rfloor, \lceil x \...
ducbadatchem's user avatar
4 votes
0 answers
242 views

Is $N=3\cdot 2^k+1$ prime if and only if $2^{N-1}\equiv 1 \pmod N$?

Is the following statement true? Let $k\geq 1$ be an integer and $N=3\cdot 2^k+1$. Then $N$ is a prime prime if and only if $2^{N-1}\equiv 1 \pmod {N}$ One implication is simply a Fermat's Little ...
Sil's user avatar
  • 17k
2 votes
1 answer
62 views

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 ...
George Plousos's user avatar
2 votes
0 answers
61 views

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}{\...
Tony Reix's user avatar
  • 393
4 votes
0 answers
99 views

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 ...
SchellerSchatten's user avatar
0 votes
1 answer
61 views

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 $\...
SeekingAMathGeekGirlfriend's user avatar
7 votes
3 answers
834 views

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.
ryan mcbean's user avatar
  • 9,389
3 votes
1 answer
136 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
53 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
  • 6,549
1 vote
0 answers
321 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
  • 6,456
0 votes
1 answer
39 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
  • 25.7k
0 votes
1 answer
100 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 &&...
user avatar
3 votes
2 answers
148 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 $\...
SeekingAMathGeekGirlfriend's user avatar
14 votes
2 answers
292 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
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