# Questions tagged [conjectures]

for question related to conjectures.

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### Asymptotics of Littlewood polynomials

Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients.
He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether ...

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### Zero divisors with support size 3 in complex group algebras of residually finite groups

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\beta$ is a non-zero element of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ such that $1\...

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**1**answer

185 views

### Zero divisors in complex group algebras of residually finite groups

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...

**5**

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**1**answer

378 views

### Is the following weak version of second Hardy-Littlewood conjecture already known?

Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the Second Hardy-Littlewood Conjecture which states that,
For all $x,y\ge 2$ we have, $$\pi(x)+\...

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**0**answers

161 views

### A conjecture regarding odd perfect numbers

(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of ...

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### Order of magnitude of extremely abundant numbers and RH

I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \...

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### Are there examples of conjectures supported by heuristic arguments that have been finally disproved?

The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...

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### Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension

In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field.
If $K$ is an imaginary quadratic field and $F/K$ is ...

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**2**answers

406 views

### Is the exponential version of Catalan-Dickson conjecture true?

The aliquot sum function $s:\mathbb{N}\rightarrow \mathbb{N}$ assigns to any natural number $n$ the sum of its proper divisors. Perfect numbers are fixed points of this function. The open conjecture ...

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**1**answer

120 views

### A rearrangement inequality for exponentiation function

Inequality: If $n$ be positive integer $n \ge 2$ and $a_1 \ge a_2 \ge \cdots \ge a_n \ge 0$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 0$ then
$${\left(\sum_{i=1}^{n}{a_i^{\...

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**2**answers

188 views

### A symmetric polynomial inequality

I improve my previous question. Because this conjecture is exactly natural development of A Muirhead Like Inequality and Muirhead's Inequality so I think the conjecture is true. But I can not prove it....

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### Does Vizing's conjecture hold for the infinite graphs?

In finite graph theory, there are many (in)equalities which relate the integer value of a certain graph invariant (e.g. domination or chromatic number) for the product of two finite graphs (e.g. ...

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### Is the conjecture A+B=C following correct?

Is the conjecture on A+B=C following correct ?
Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write:
...

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**3**answers

216 views

### Density of the Klarner-Rado Sequence

Consider the Klarner-Rado sequence OEIS A005658 defined by the rule: the sequence starts with 1, and if it contains $n$ it also contains $2n$, $3n+2$ and $6n+3$. According to R. Guy's popular article,...

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**1**answer

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### Relative to Isoperimetric inequality with n-polygon

Since Brahmagupta's formula and Bretschneider's formula we have the inequality:
Any two quardrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ with the same sidelengths and $A_1A_2A_3A_4$ is a cyclic ...

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**1**answer

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### Strengthened version of Isoperimetric inequality with n-polygon

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...

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### The Krzyż Conjecture

What is the state of the Krzyż Conjecture? It states for that for all $f:\mathbb{D}\to \bar{\mathbb{D}}$ holomorphic and non-vanishing, the coefficients $a_n$ in the power series of $f$ are at most $2/...

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**1**answer

171 views

### Discriminant of numerator of inverse logarithmic derivative operator iteration

Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$Tf=\frac{f}{f'}.$$ Define $$p_n(x)=T^n\left(x-\frac{x^2}{2}\right).$$ Let $f_n(x) \in \mathbb Z[x]$ be ...

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**1**answer

210 views

### Find all postive integer$n$ such $n+\tau{(n)}=2\varphi{(n)}$

conjecture：Today I have no intention of thinking about this question. I have only got two solutions so far. I guess there are only two solutions, but I won't prove it.
Let $n$ be postive integers,...

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**1**answer

4k views

### Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?

In this question on MSE, Enzo Creti asks for a prime number formed by concatenating the Mersenne numbers $2^n-1$ and $2^{n-1}-1$, for example, 40952047. For all residues modulo 7, he found primes ...

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### Inequality in Frankl's conjecture

For the minimal counter-example to union closed sets conjecture, we have the lower bound $\mid$$\mathcal{A}$$\mid$ $\geq$ $4q-1$ ($\mathcal{A}$ denotes the minimal counter-example family, $q$ denotes ...

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### Mathematical conjectures on which applications depend

What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?

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**1**answer

170 views

### The difference between two coprime semiprimes

Conjecture:
Any positive integer can be written as the difference between two
coprime semiprimes.
Tested up to 1,000,000.
See also:
https://math.stackexchange.com/questions/2579578/the-...

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**0**answers

88 views

### Common insights into hypothetical complexity of graph problems

I came across two examples of hypothetical hardness of some graph problems. Hypothetical hardness means that refuting some conjecture would imply the NP-completeness of the respective graph problem. ...

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**2**answers

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### A conjecture regarding prime numbers

For $n,m \geq 3$, define $ P_n = \{ p : p$ is a prime such that $ p\leq n$ and $ p \nmid n \}$ .
For example :
$P_3= \{ 2 \}$
$P_4= \{ 3 \}$
$P_5= \{ 2, 3 \}$,
$P_6= \{ 5 \}$ and so on.
Claim: $...

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votes

**1**answer

172 views

### Two questions on “Table problem on $\Bbb S^2$”

The following conjecture is known as "Table problem on $\Bbb S^2$"
Conjecture (Table problem on $\Bbb S^2$): Suppose $x_1, x_2,x_3,x_4 \in\Bbb S^2 \subseteq \Bbb R^3$ are the vertices of a
square ...

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**1**answer

183 views

### Can one find a Jordan curve which has exactly one inscribed rectangle?

In On the number of inscribed squares of a simple closed curve in the plane it is shown that
Theorem: For every positive integer $n$ there is a simple closed curve in the
plane (which can be ...

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**1**answer

196 views

### Has the Total Coloring Conjecture been proved for complete graphs?

I have a question on the Total Coloring Conjecture in graph theory. This conjecture states that
$$\chi^"(G)\leq \Delta +2,$$
where $\Delta$ is the maximum degree of the graph and $\chi^"(G)$ denotes ...

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**1**answer

1k views

### Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$

Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+...

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191 views

### Does this idea give an algorithm for constructing Hadamard matrices?

Fedor Petrov's answer of my preceding question shows that my question reduces to the famous Hadamard conjecture about Hadamard matrices of order $4k$. So I decided to study this conjecture and I got ...

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208 views

### The set of numbers $a+b$ such that $ma^2+nb^2$ is prime

Conjecture:
If $m,n$ are coprime it exist a minimal natural number $N_{mn}$ such
that:
$\{a+b>N_{mn}\mid a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P_{>2}\} = \{ k > N_{mn} \mid \...

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### Testing the Cartan determinant conjecture via Gorenstein algebras

Let $A$ be a Gorenstein algebra (of infinite global dimension) with finitely many indecomposable Gorenstein projective modules and $X$ the basic direct sum of all indecomposable Gorenstein projective ...

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**2**answers

233 views

### Sylvester–Gallai theorem with circle version, plane version and curve version?

The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either
All the points are collinear; or
There is a line which contains exactly two of the ...

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**1**answer

267 views

### Solve this diophantine equation: $m^4+n^4=10m^2n^2+1$

t's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo nn for every nn. This fact is stated, for example,
...

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**1**answer

440 views

### Which motivic cohomology groups of complex numbers are non-torsion?

I would like to know which motivic cohomology groups of complex numbers are non-zero and ("better") non-torsion, i.e., for which $(i,j)$ the $i$th cohomology of the complex ${\mathbb{Q}}(j)$ over $\...

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**0**answers

118 views

### Is there a composite number for which this game with composite numbers never ends?

I created some "game" a couple of days ago and modified it during the last hour or so and now I want to present it to you.
Choose some natural number $>1$. If the chosen natural number is of the ...

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**0**answers

97 views

### Question in the setting of generalized Diophantine $m$-tuples

As an amateur I am not quite sure should I post a question on the site for professional mathematicians but if the question is not appropriate for this site you can freely migrate it to ...

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### Dirac's theorem and the 1-factorization conjecture

Let $G=(V,E)$ be a simple, undirected graph. A matching is a subset $M\subseteq E$ such that all members of $M$ are pairwise disjoint; moreover we call $M$ perfect if $\bigcup M = V$.
The 1-...

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**2**answers

319 views

### Generalization of Legendre`s conjecture

Legendre`s conjecture states that there is always a prime between $n^2$ and $(n+1)^2$ for every natural $n$.
It is natural to create following generalization:
Is it true that for every $\...

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votes

**2**answers

297 views

### Maximum matching in a graph with no “shortcuts”

For a directed acyclic graph (DAG) $G$, denote by $G^\star$ the undirected graph obtained from $G$ by ignoring direction of its arcs. Let $e(G)=e(G^\star)$ be the number of arcs in $G$ (or edges in $G^...

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**0**answers

77 views

### Primes dividing functions defined by linear recurrence relations with constant coefficients

For Fibonacci numbers $F_n$ it holds that $p|F_{p-(\frac{5}{p})}$, if $p$ is an odd prime (Legendre symbol).
I guessed that the number $5$ came from the roots of the characteristic polynomial and ...

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**1**answer

427 views

### A conjecture about certain values of the Fabius function

The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations
$$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$
and
$$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\...

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**1**answer

457 views

### Several conjectured identities for polylogarithms

I asked a question at M.SE a couple of years ago about polylogarithms $\!^{[1]}$$\!^{[2]}$$\!^{[3]}$$\!^{[4]}$ where I conjectured
$$720\,\operatorname{Li}_4\!\left(\tfrac12\right)-2160\,\operatorname{...

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**1**answer

621 views

### What is the status on this conjecture on arithmetic progressions of primes?

The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.
For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of ...

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**3**answers

668 views

### A curious series related to the asymptotic behavior of the tetration

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence
$$
{^{-1} a} = 0, \quad {^{n+1} a} = a^{\...

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**4**answers

1k views

### A seemingly simple inequality

Let $a_i,b_i\in\mathbb{R}$ and $n>1$, does the inequality
$$
\left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)+\left(\sum_{i=1}^na_i b_i\right)^2\ge \sqrt{\left(\sum_{i=1}^n a_i^4\...

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**1**answer

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### Reconstructing analytic tetration with a complex height from a thinner set of points

This is a follow-up to my previous question An explicit series representation for the analytic tetration with complex height.
Recall the definition $(11)$ from there:
$$t(z) = \sum_{n=0}^\infty \sum_{...

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**3**answers

1k views

### An explicit series representation for the analytic tetration with complex height

Tetration is the next hyperoperation after more familiar addition, multiplication and exponentiation. It can be seen as a repeated exponentiation, similar to how exponentiation can be seen as a ...

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**1**answer

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### Solve an equation with a factorial [closed]

x! - 2 = y^2.
Task: Solve over the naturals.
I think the answers are x = 3, x = 2, but I am not sure.

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**1**answer

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### Proving $k = 1 \implies q = 5$, if $q^k n^2$ is an odd perfect number with Euler prime $q$

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$.
We want to show that the biconditional $k = 1 \iff q = 5$ holds.
It suffices to prove one direction, as the implication $q = 5 \...