# Questions tagged [conjectures]

for question related to conjectures.

142
questions

**-2**

votes

**1**answer

181 views

### A conjecture stronger than the Legendre conjecture about prime numbers [closed]

I want to draw attention on my own question (cross posted from Mse)
Hi I want to evaluate the following sum :
$$S_n=\frac{1}{\log(2)}+\frac{2}{\log(3)}+\frac{3}{\log(4)}+\frac{4}{\log(5)}+\cdots+\...

**4**

votes

**0**answers

152 views

### A generalization of Feit–Thompson conjecture, for square-free integers

I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...

**4**

votes

**1**answer

221 views

### About semiprimal representations of $1$

Conjecture $A_1$ : For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ and numbers $i_1,...,i_m \...

**0**

votes

**1**answer

365 views

### Has Pillai conjecture been proven?

I found the paper https://hal.archives-ouvertes.fr/hal-00698687v9/document which claims the proof of Pillai conjecture.
However, it is not mentioned anywhere that it has been proved. It's stated ...

**2**

votes

**0**answers

331 views

### Equivalence of the union-closed sets conjecture that is locally weaker of any use?

Let $F$ be a union-closed family. We call $F$ minimal if for every $x\in \cup(F)$ we find $S\in F$ such that $S\backslash \{x\} \in F$. It is sufficient to proof the union-closed sets conjecture for ...

**2**

votes

**0**answers

48 views

### Schur positive expression involving border-strip tableaux

Recall the power-sum expansion of Schur functions,
$$
s_\lambda = \sum_\mu \chi^{\lambda}(\mu) \frac{p_\mu}{z_\mu},
$$
in terms of Sn-character. These can be calculated
by the Murnaghan-Nakayama rule, ...

**7**

votes

**1**answer

331 views

### Conjectured primality test for specific class of $N=4kp^n+1$

Can you provide a proof or counterexample for the following claim?
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $...

**5**

votes

**2**answers

532 views

### Conjectures and open problems in representation theory [closed]

Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?

**0**

votes

**0**answers

79 views

### Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function

The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...

**5**

votes

**0**answers

566 views

### is there a link with the probabilistic model for prime numbers?

Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...

**14**

votes

**1**answer

514 views

### Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.
We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...

**-3**

votes

**1**answer

323 views

### Published articles in journals about the Firoozbakht's conjecture, whose main goal or focus is the study of this conjecture

I would like to know what articles are in the literature about the known as Firoozbakht's conjecture, see the Wikipedia Firoozbakht's conjecture.
Question. What articles have been published in ...

**2**

votes

**1**answer

123 views

### On $\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{(\log p)^m}{p}$, on assumption of the first Hardy–Littlewood conjecture

I wondered, inspired in a result from [1] (Proposition 17) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture,
$$\sum_{\substack{\text{...

**-1**

votes

**1**answer

220 views

### Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]

$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$.
If the ...

**1**

vote

**0**answers

37 views

### On quasiperfect numbers, $\operatorname{rad}(\sigma(n))=\operatorname{rad}(2n+1)$ and $\operatorname{rad}(\sigma(n)-1)=\operatorname{rad}(2n)$

An integer $n\geq 1$ is said quasiperfect number if the sum of its positive divisors $\sigma(n)$ is equal to $2n+1$. See the Wikipedia Quasiperfect number.
The idea of this post is ask about the ...

**2**

votes

**1**answer

340 views

### A generalization of Lander, Parkin, and Selfridge conjecture

My question: Are the conjectures as follows correct?
Given a positive integer $P>1$, let its prime factorization be written
$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$.
Define the ...

**0**

votes

**0**answers

60 views

### Inequalities (or other statments) that satisfies the twin prime constant on assumption of Nicolas equivalence to the Riemann hypothesis

The idea of this post is try to set some statement that involves the twin prime constant $\Pi_2$ and Nicolas criterion for the Riemann hypothesis. I wrote a statement merely as a simple combination of ...

**3**

votes

**2**answers

637 views

### Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...

**1**

vote

**0**answers

179 views

### Is it possible to get an interesting statement about even perfect numbers from the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$?

It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...

**0**

votes

**1**answer

82 views

### An triangle inequality $\sum_{i=1}^n b_i^\alpha \ge \sum_{i=1}^na_i^\alpha $ if $\alpha >1$

Using my computer I discovered that:
if $a,b,c$ are sidelengths of a triangle, then
$(a+b-c)^\alpha+(b+c-a)^\alpha+(c+a-b)^\alpha \ge a^\alpha+b^\alpha+c^\alpha $ if $\alpha >1$
$(a+b-c)^\alpha+...

**53**

votes

**5**answers

5k views

### Arriving at the same result with the opposite hypotheses

I am pretty distant from anything analytic, including analytic number theory but I decided to read the Wikipedia page on the Riemann hypothesis (current revision) and there is some pretty interesting ...

**9**

votes

**0**answers

392 views

### Why to believe the Fargues geometrization conjecture?

In the study of the arithmetic local Langlands correspondence, there is a conjecture that was recently (in this decade) formulated by Fargues.
I can't even concisely state the conjecture so I will ...

**5**

votes

**5**answers

223 views

### Peculiarities in low dimensions or low order or etc

I have been pondering about certain conjectures and theorems viewed as either low vs high dimensions, or smaller vs larger primes, or anything of the sort "low vs high order". Let me mention a couple ...

**8**

votes

**0**answers

389 views

### Seeking proof of the Cuckoo Cycle Conjecture

Allow me to introduce the Cuckoo Cycle Proof of Work, and conclude with a closely related Conjecture about random graphs, which I hope someone can find a proof for.
Cuckoo Cycle is named after the ...

**9**

votes

**0**answers

250 views

### Symmetric function transition matrix and a non-conjecture by Clifford and Stanley

Consider the transition matrix $R = \left(R_{\lambda,\mu}\right)$, defined by
$$
p_\lambda = \sum_{\mu} R_{\lambda\mu}m_\mu ,
$$
between the power-sum and the monomial basis of the ring of symmetric ...

**19**

votes

**4**answers

1k views

### Can anything deep be said uniformly about conjectures like Goldbach's?

This is a soft question sparked by my curiosity about the intrinsic depth of Goldbach-like conjectures as perceived by current experts in number theory. The incompleteness theorem implies that, if our ...

**4**

votes

**1**answer

279 views

### Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

This question is an old question from mathstackexchange.
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that
$$\sup f_-...

**1**

vote

**0**answers

108 views

### Testing polynomials irreducible over the integers

Let $f\in\operatorname{int}(\mathbb Z)$, the ring of integer-valued rational polynomials. Define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments....

**2**

votes

**1**answer

213 views

### Integer valued polynomials over several variables

For simplicity this is about polynomials in just two variables.
Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials
$X^iY^j$ and therefore as a sum of polynomials $p_{ij}\...

**8**

votes

**1**answer

2k views

### Percentage of Ramanujan's conjectures that were proven correct

Today I read the following brief but insightful account of Ramanujan's approach to mathematics: https://www.imsc.res.in/~rao/ramanujan/images/KSRchap3.pdf and while reading this I wondered whether we ...

**15**

votes

**2**answers

436 views

### A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture

The Rohrlich-Lang Conjecture for polynomial relations in Gamma values predicts that all polynomial relations between Gamma values over $\mathbb Q$ come from the functional equations satisﬁed by the ...

**2**

votes

**0**answers

259 views

### 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 ...

**3**

votes

**0**answers

54 views

### 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\...

**6**

votes

**1**answer

242 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**

votes

**1**answer

400 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)+\...

**4**

votes

**1**answer

380 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 ...

**8**

votes

**0**answers

239 views

### 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 \...

**47**

votes

**6**answers

4k views

### 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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**0**answers

173 views

### 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 ...

**2**

votes

**2**answers

421 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 ...

**3**

votes

**3**answers

263 views

### A rearrangement inequality for exponentiation function

Update: A year ago, but the first answer is not clear with me. I bounty this question again.
My question: I am looking for a proof or counterexample to the following inequality:
If $n \in \mathbb{...

**0**

votes

**2**answers

230 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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votes

**1**answer

405 views

### 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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votes

**0**answers

4k views

### 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:
...

**8**

votes

**3**answers

296 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,...

**3**

votes

**1**answer

110 views

### 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 ...

**9**

votes

**1**answer

421 views

### 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}}\...

**5**

votes

**0**answers

231 views

### 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/...

**7**

votes

**1**answer

188 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 ...

**3**

votes

**1**answer

235 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,...