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

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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} ...
3
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1answer
47 views

A relation related with odd perfect numbers

It is easy to prove, using the relation $\prod_{d\mid n}d=n^{\sigma_0(n)/2}$ holds for $n\geq 1$ where $\sigma_0(n)$ is the number of divisors, the following Proposition. The integer $n\geq 1$ is ...
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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 ...
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2answers
40 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 ...
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1answer
21 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 ...
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0answers
122 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 ...
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0answers
26 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 ...
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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 ...
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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 $p=13,17,19,23$ ...
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95 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 ...
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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 ...
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0answers
128 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 $1$. ...
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12answers
10k views

Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a ...
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0answers
29 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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9answers
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Why do mathematicians sometimes assume famous conjectures in their research?

I will use a specific example, but I mean in general. I went to a number theory conference and I saw one thing that surprised me: Nearly half the talks began with "Assuming the generalized Riemann ...
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2answers
167 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. ...
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0answers
66 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 ...
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1answer
617 views

The Goldbach Conjecture and Hardy-Littlewood Asymptotic

A source I am reading refers to the Goldbach conjecture (that every even number is the sum of two primes), and then immediately follows with the "Hardy-Littlewood conjecture" that $\sum ...
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1answer
83 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?
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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$. ...
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3answers
206 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 ...
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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 ...
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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 $3$. ...
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1answer
366 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 ...
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2answers
529 views

Why Goldbach's conjecture is difficult to prove?

Why Goldbach's conjecture is still non-solved and is difficult to prove? What makes the mathematicians fail when trying to prove it?
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0answers
210 views

Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
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1answer
159 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 ...
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4answers
726 views

Closed form for $\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx$

I'm trying to find a closed form for the following integral: $$\mathcal{J}(n)=\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx\tag1$$ I have conjectured values ...
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2answers
292 views

Yet another conjecture about primes

Let $\mathcal{N}(n)$ be the next prime greater than $n$. Conjecture: $\mathcal{N}(n!)-n!\:$ is either $1$ or a prime. It holds for n=1 to 99 and the expression is 1 for 3,11,27,37,41,73,77 and ...
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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 $\Delta:\mathbb N\times\mathbb N\to\mathbb N$. ...
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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 ...
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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 ...
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1answer
98 views

How to convert an object into a sphere?

I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a ...
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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
163 views

Firoozbakht's conjecture and maximal gaps

In the Wikipedia article, it seems to me as if it's implied that it is enough to check the conjecture only for maximal gaps (numbers $n$ s.t. $\forall k<n:g_n>g_k$). I.e it holds that ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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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85 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), $C_2$ ...
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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} ...
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2answers
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Does giving a counterexample to a conjecture prove it to be true or false?

If a problem asks me to give a counterexample to a conjecture, am I proving the conjecture true or false by giving a counterexample? I am leening towards proving it false, because if I were to prove ...
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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
87 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 ...
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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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332 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 ...
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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 ...