Use this tag if your question is about a well-known conjecture or a conjecture of your own.

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20 views

Is always two times an even semiprime at a distance $1$ or prime to the closest previous odd semiprime?

This is an observation regarding the semiprimes, also named 2-almost primes, biprimes, or the product of two primes. This week I do not have a computer, only a tablet (hospitalized with a lot of free ...
2
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1answer
30 views

Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
2
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1answer
43 views

Conjecture: $\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$

I conjecture: $$\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$$ Is this provable? If this is false, then can I have a function $f$ such that: $$\lim_{x\to\infty} ...
0
votes
2answers
41 views

Special case of Pillai's conjecture

Pillai's conjecture is a generalization of Catalan's conjecture. It's say that for fixed positive integers $A, B, C$ the equation $Ax^n - By^m = C$ has only finitely many solutions $(x,y,m,n)$ with ...
0
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1answer
24 views

Rational solutions for $\sin(n)$ in radians

This is completely for my own curiosity. Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians. I know that this is strange because usually integers are only used in ...
5
votes
1answer
22 views

Zauner's conjecture

The conjecture is as follow: In $\mathbb{C}^{n}$, there exists $\{v_1,\cdots,v_{n^2}\}$ such that the following holds: $$ \left| \left \langle v_i, v_j \right \rangle \right| = \begin{cases} 1 ...
1
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0answers
27 views

Does the antipode of a f.d. Hopf algebra have finite order

In a lecture I have heard that the antipode of a finite-dimensional Hopf algebra is conjectured to be finite, and that this has only been proven in characteristic $0$ by Larson and Radford in 1988. I ...
4
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4answers
135 views

Group conjecture

Conjecture: Given a finite group $G$ and a subset $A\subset G$. Then $\{A,A^2,A^3,\dots\}$ is a group iff $\forall n\in \mathbb N: |A^n|=|A^{n+1}|$. Given that the composition between the ...
3
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0answers
96 views

Combining Firoozbakht's conjecture and abc conjecture

Firoozbakht's conjecture states that for all $n\geq 1$ $$p_n^{\frac{1}{n}}>p_{n+1}^{\frac{1}{n+1}},$$ where $p_k$ the kth prime number. By asumption of this conjecture, for a fixed $n$, there is a ...
2
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1answer
145 views

Is there a typo in the formula or does my GAP-package sglppow fail?

This site deals with group formulas for prime powers $p^k$ for $k\le 7$. The formula for $k=7$ seems to be wrong. I compared the results with GAP and the formula is off by $2453$ for $p=13,17,19,23$ ...
0
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0answers
30 views

What's the biggest number used as a counterexemple? [duplicate]

I'm looking for exemples of big numbers that are counterexemple of some interesting conjecture. Do you know conjectures that seemed to be true until a million (or many more) numbers where checked?
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0answers
151 views

Conjecture $\sum_{n=1}^\infty\frac{(1/2)_n(-1/6)_n}{n!(2/3)_n}H_n\overset{?}={\pi\over 6}+2\sqrt{3}\ln(1+\sqrt{3})-{7\over\sqrt{3}}\ln 2-6+2\sqrt{3}$

Sums involving harmonic numbers have old history and first examples of their evaluations go back to Euler. Lately there has been a lot of interest, both on this forum and among professional ...
0
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0answers
130 views

Kolmogorov 0-1 Law Conjecture

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. ...
2
votes
0answers
67 views

Fibonacci Numbers and the Harmonic Series

$$\sum_{k=1}^{n} \frac{1}{k}=H_n=\frac{p_n}{q_n}$$ Where $p_n,q_n$ are coprime intergers. The first few values for $p_n+q_n$ are $2,5,7,37,197,69,504,1041,9649$. When are $p_n+q_n$ Fibonacci ...
8
votes
2answers
170 views

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
5
votes
1answer
84 views

Why Riemann hypothesis and not Riemann's conjecture

I have a stupid question. We say Erdös's conjecture, Goldbach's conjecture, Beal's conjecture... and so on. But we don't say 'Riemann's conjecture.' Instead we use the word 'hypothesis'. Why?
5
votes
1answer
46 views

Recreational conjecture on factoring groups

Consider the following: For a group $G$ with identity $e$, define $s: G \to \mathbb{N} \cup \{ \infty \}$ by $s(g) = \min \{ k \in \mathbb{N} : g^{k} = e \}$, where $ \min \emptyset = \infty$. ...
2
votes
2answers
83 views

Sum and product of k real numbers > 0 is unique?

Can we prove that $\sum_{i=1}^k x_i$ and $\prod_{i=1}^k x_i$ is unique for $x_i \in R > 0$? I stated that conjecture to solve CS task, but I do not know how to prove it (or stop using it if it is ...
2
votes
1answer
31 views

Conjecture for the maximum number of rooms.

The puzzle I have is essentially this, but for $n$ rows.For this instance of $n=5$, quick tallying reveals the answer to be $21$. For $n=4$, it is $13$. For $n=3$, it is $7$. For $n=2$ it is $3$. ...
13
votes
3answers
207 views

Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$

While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$ S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3}, $$ where ...
5
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0answers
125 views

Are all totients of Fibonacci Numbers distinct?

This question was inspired while I was seeing how certain recurrence relations would behave when I applied Multiplative Functions. Let $F_{n}$ be a sequence for which $F_{1}=1,F_{2}=1$, and ...
2
votes
1answer
70 views

A conjecture about the prime counting function

Using this lemma it can be proved that $\Delta(m,n)=\pi(m\cdot n)-\pi(m)\cdot\pi(n)+1$ (where $\pi$ is the prime counting function) is a function $\Delta:\mathbb N\times\mathbb N\to\mathbb N$. ...
6
votes
1answer
160 views

A conjecture about prime numbers based on $\sigma_1(n)$ and the Highly Abundant Numbers

I am trying to find the smallest expression $E(n)$, whose distances between the value of the expression and the next prime closer to the expression, $\mathcal{N}(E(n))$, and from the expression to the ...
14
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1answer
373 views

The four runner problem/conjecture

I've recently read here the following problem, called « four-runner problem » : Suppose four runners (represented by labeled spheres) run around a circular track. Their speeds are constant ...
1
vote
1answer
38 views

Among any $2n$ consecutive integers below $n^2+2n$ at least one has no prime divisor less than $n$?

What is your idea about my conjecture? Consider a sequence of $2n$ consecutive natural numbers, all the terms less than $n^2 + 2 n$. Then there exists at least one number in the sequence which is ...
1
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0answers
38 views

Sketch of a possible equivalence with Riemann hypothesis

From Robin's equivalence (see [1]) and the following trigonometrics identitites, I ask to me if it is feasible write vagues equivalences using this strong result and if these equivalences will be ...
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0answers
24 views

What are equivalent forms of Collatz conjecture in different disciplines of mathematics?

My question is about known equivalent forms of Collatz conjecture in different fields of mathematics. Any reference to a similar conjecture or theorem in these fields is also welcome.
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2answers
34 views

With $s(n)=\sum_{k=1}^n n \bmod k$, can be justified that $\forall\epsilon>0$ let us $\lim_{n\to\infty}\frac{s(n-1)}{\epsilon+s(n)}=1?$

Denoting as $$s(n)=\sum_{k=1}^n n \bmod k$$ the sum of remainders function (each remainder is defined as in the euclidean division of integers $n\geq 1$ and $k$). See [1] for example. For examples ...
0
votes
1answer
68 views

Jacobian conjecture over $\mathbb C$ and over any field of characteristic zero

The well known Jacobian conjecture is Jacobian Conjecture. If $F:\mathbb C^n\rightarrow\mathbb C^n$ is a polynomial map and the Jacobian matrix $J(F)$ is invertible, then $F$ is an invertible ...
2
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0answers
64 views

Twin primes sums conjecture

I have found an interesting conjecture between twin primes sums. I don't know if it is already described by someone else. I have checked in internet, but I didn't find any mention of such conjecture. ...
1
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0answers
35 views

If $\gcd(Z,\sigma(Z))=1$ and $N=Z\sigma(Z)$, is $N$ always friendly?

This question is a generalization / offshoot of this earlier MSE post: If $\gcd(Z,\sigma(Z))=1$ and $1<N=Z\sigma(Z)$, is $N$ always friendly? Here, $\gcd(a,b)$ is the greatest common divisor ...
1
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0answers
79 views

Why is proving that $10$ is solitary considered very difficult?

The title says it all. We denote the sum of the divisors of $x$ by $\sigma(x)$. The ratio $I(x)=\sigma(x)/x$ is called the abundancy index of $x$. If $I(m)=I(n)$, then $\{m,n\}$ is called a ...
1
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0answers
41 views

Is $p(p + 1)$ always a friendly number for $p$ a prime number?

Let $\sigma(x)$ denote the sum of the divisors of $x$. We call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A positive integer $N$ is friendly if there exists another positive ...
4
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1answer
56 views

Which heuristic leads to the Hardy-Littlewood conjecture about twin primes?

According to Wikpedia, Hardy-Littlewood conjecture says that $$\pi_2(n) \sim 2 C_2 \frac{n}{(\ln n)^2} \sim 2 C_2 \int_2^n {dt \over (\ln t)^2}$$ where $$C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} ...
3
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1answer
89 views

Twin prime conjecture implies $\limsup_{n\to\infty}\frac{\sigma(n)\pi(n)}{n^2}\left(\pi(\log n)-\frac{\pi_2(\log n)}{2C_2}\right)=e^{\gamma}$?

Let $\sigma(n)$ the sum of positive divisor function, $\pi(x)$ is the prime counting function, $\pi_2(x)$ is the twin prime counting function (we will assume that Twin prime conjecture holds), $C_2$ ...
10
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1answer
88 views

Is it true/known/important that $(\log p_n)/n$ is nonincreasing, where $p_n$ is the $n$th odd prime number?

First thing first, I would like to apologize in advance for my poor knowledge of Maths and English. I'm an Italian student and after asking to all the mathematicians and Maths teachers in my town, I ...
1
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1answer
52 views

a conjecture of two equivalent q-continued fractions related to the reciprocal of the Göllnitz-Gordon continued fraction A111374-OEIS

Given the square of the nome $q=e^{2i\pi\tau}$ and ramanujan theta function $f(a,b)=\sum_{k=-\infty}^{\infty}a^{k(k+1)/2}b^{k(k-1)/2}$ with $|q|\lt1$, define, ...
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1answer
34 views

Iterated $\sigma^k(n)$, excluding the possibility that primes $p|\sigma^t(n)$, $t< k$ divides an odd perfect number $n$

Let $m\geq 1$ an integer and $\sigma(m)=\sum_{d|m}$ the sum of positive divisors function. A positive integer is said to be perfect if and only if $\sigma(n)=2n$. We have that $\sigma(m)$ is a ...
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2answers
36 views

4 Divides x Proofs of conjectures

Hi there I'm working on a set of problems and I'm having some difficulty proving and disproving these examples. I know that #1 is essentially (There exists K where [x=4k]) I'm lost after that. I'm not ...
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0answers
24 views

Criteria for almost perfect and $p$-deficient numbers

Let $\sigma$ be the (classical) sum-of-divisors function. A number $n$ is called almost perfect if $\sigma(n) = 2n - 1$. If $\sigma(m) = 2m - p$ for some integer $p > 1$, then $m$ is called ...
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0answers
34 views

Number of Magic squares and their applications

A magic square is a square array of numbers consisting of the distinct positive integers $1, 2, ..., n^2$ arranged such that the sum of the $n$ numbers in any horizontal, vertical, or main diagonal ...
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4answers
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Is the 3x+1 problem solved? [closed]

I found an article by Peter Schorer from June 29,2015 which is claming to give a solution of the 3x+1 problem. Are there remarks from any mathematicians if this is correct or not?
5
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1answer
56 views

An asymptotic behavior of $\operatorname{Li}_{-n}(a)$ for $n\to\infty$

Suppose $a,b\in(0,1)$. I'm interested in comparison of an asymptotic behavior of $\operatorname{Li}_{-n}(a)$ and $\operatorname{Li}_{-n}(b)$ for $n\to\infty$. Such functions exhibit approximately ...
6
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0answers
118 views

Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us. Using ...
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0answers
45 views

Does the following equation have any solutions in $\mathbb{N}$?

Let $\mathbb{N}$ be the set of positive integers. The function $\sigma(N)$ gives the sum of the divisors of $N$. My question is: Does the following equation have any solutions for $x \in ...
3
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0answers
62 views

Up to which value is Rassias' conjecture verified?

I came across this conjecture: Rassias' conjecture Up to which $p$ has this conjecture be verified ? Are there intermediate results related to this conjecture ? The conjecture can be ...
2
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1answer
36 views

Proving the existence of consecutive quadratic residues modulo $p>5$ by means of Pell's equation

When I read Existence of Consecutive Quadratic residues I thought we might be able to prove the existence of non-zero consecutive quadratic residues modulo a prime $p>5$ using the theory of ...
3
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0answers
47 views

Why Navier-Stokes solutions are expected to be $C^\infty$?

I am curious, why the Millennium problem about Navier-Stokes is about smooth ($C^\infty$) not just $C^1$ solutions?
9
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1answer
333 views

a conjectured continued-fraction for $\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$

In this post,I posed a similar conjecture for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$ but did not get any helpful answers. Given a complex number ...
0
votes
1answer
28 views

about conjecture “every finite nonabelian p-group admits a noninner automorphism of order p”

by "The Kourovka Notebook. Unsolved Problems in Group Theory" there is a strong conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. What about finite ...