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for question related to conjectures.

2
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0answers
148 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
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0answers
225 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 \...
45
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6answers
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 ...
1
vote
0answers
61 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 ...
3
votes
2answers
336 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 ...
0
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1answer
117 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^{\...
0
votes
2answers
175 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....
8
votes
1answer
377 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. ...
11
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0answers
3k 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
3answers
211 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
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1answer
77 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
1answer
370 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
0answers
127 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
1answer
170 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
1answer
208 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,...
1
vote
1answer
3k 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 ...
1
vote
0answers
132 views

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 ...
65
votes
9answers
7k views

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?
1
vote
1answer
158 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-...
3
votes
0answers
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. ...
54
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2answers
3k views

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: $...
4
votes
1answer
171 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 ...
3
votes
1answer
178 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 ...
2
votes
1answer
181 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 ...
18
votes
1answer
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+...
1
vote
0answers
187 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 ...
6
votes
0answers
206 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 \...
2
votes
0answers
58 views

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 ...
8
votes
2answers
230 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 ...
1
vote
1answer
265 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, ...
16
votes
1answer
418 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 $\...
8
votes
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
86 views

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-...
5
votes
2answers
318 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 $\...
8
votes
2answers
296 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^...
1
vote
0answers
76 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 ...
10
votes
1answer
421 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.\...
14
votes
1answer
455 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{...
21
votes
1answer
608 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 ...
16
votes
3answers
661 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^{\...
18
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4answers
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\...
3
votes
1answer
131 views

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_{...
19
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3answers
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 ...
-2
votes
1answer
143 views

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.
0
votes
1answer
122 views

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 \...
4
votes
0answers
290 views

there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

Let $S_n:=1+\frac12+\frac13+\ldots+\frac1n$. Is it true that the set of $n\in\mathbb N$ such that $$S_n-[S_n]<\dfrac{1}{n^2}$$ is infinite? Here, $[x]$ represents the largest integer not exceeding $...
59
votes
7answers
7k views

Theorems demoted back to conjectures

Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered. I am ...
13
votes
2answers
605 views

A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol: $$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$ Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes ...
3
votes
0answers
194 views

Any counter example for this: ${\phi(2^n-1)} \bmod \tau(2^n-1)=0$ for every integer $n \geq 1$? [closed]

I asked this question here In S.E but i don't received any resposnes for it, I would like to know if it is appropriate for M.O. I'm always interesting for properties of the following series : $ \...