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

learn more… | top users | synonyms

8
votes
0answers
130 views

Mathematical conjectures for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...
0
votes
0answers
10 views

Unsolved problems involving or concerning the unitary DFT

Does anyone know of any unsolved problems involving or concerning the unitary discrete Fourier transform matrix $F_n=n^{-1/2}(f_{j k})$ where $f_{jk }=e^{2\pi j k i}$ and $i=\sqrt{-1}$, or its ...
2
votes
1answer
53 views

Explanation for relationship between hypotenuse segments and leg lengths?

|` | ` x | ` | ` c a | z /` | / ` y |_/ ` |/|_____` b I'm new to this SE, but I have an SO account, so hello! Assume ...
2
votes
0answers
60 views
+50

A conjecture about the prime function $p_n$

While testing my system Zet for computational mathematics I find possible relations now and then. The latest is: Conjecture: For all $(m,n)\in\mathbb Z_+^2$ except $(3,4),(4,3) \text{ and } ...
20
votes
6answers
2k views

What does proving the Collatz Conjecture entail?

From the get go: i'm not trying to prove the Collatz Conjecture where hundreds of smarter people have failed. I'm just curious. I'm wondering where one would have to start in proving the Collatz ...
0
votes
1answer
51 views

Collatz Conjecture, sufficient to show odd numbers reach $1$?

The famous conjecture: Let $$ f(n) = \begin{cases} n/2 & \quad \text{if } n \text{ is even}\\ 3n+1 & \quad \text{if } n \text{ is odd}\\ \end{cases} $$ The Collatz Conjecture ...
3
votes
1answer
200 views

conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
0
votes
1answer
32 views

poincare conjecture understanding

I have knowledge of basic school math and in colleges I have read calculus(mostly forgotten now). I need to understand poincare conjecture and hence I need to study a lot of things. I need to know ...
2
votes
0answers
124 views

A conjecture about primes

Let $p_n$ be the $nth$ prime and define $p_n^{(m)}$ by $p_n^{(1)}=p_n$ and $p_n^{(m+1)}=p_{p_n^{(m)}}$: $p_n^{(2)}=p_{p_n}$, $\;p_n^{(3)}=p_{p_{p_n}}$ and so far... For some coprime numbers $a,b$, ...
81
votes
12answers
12k 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 ...
1
vote
0answers
20 views

Factorial and primorial twin primes

Factorial primes are are primes of the form $n! \pm 1$ and primorial primes are primes of the form $p\#\pm 1$, where $p\#$ is the product of all primes $\leq p$. To cite ...
2
votes
0answers
38 views

Number of primes of a certain form

Let $p_n$ be the nth prime. Are there an infinite number of primes of the form $2p_n+1$? Is something known about questions like this?
8
votes
1answer
133 views

Possibly New Prime Conjecture

I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows: Any given sum of twin primes ...
1
vote
0answers
52 views

$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

Conjecture For $n \ge 1 $ , $m \ge 1$ $\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$ where $\pi\left(n\right)$ is the prime counting function . Does this conjecture ...
59
votes
2answers
2k views

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. ...
1
vote
1answer
70 views

Do all primes occur in some sequence associated with the Collatz conjecture?

Let $f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases}$ For an arbitrary prime $p$ are there some start value $x_0$ such that $p = x_k$ for some ...
3
votes
1answer
47 views

Conjecture about odd primes

For each odd prime $p$ there exist $n\in\mathbb{N}$ such that $p\equiv n^2 \text{ (mod }\varphi(n^2))$, where $\varphi$ is Euler's totient function. I'm developing my Forth based computational ...
2
votes
2answers
48 views

Factoring the factorials

Just for the fun of it, I've started factoring $n!$ into its prime divisors, and this is what I got for $2\leq n\leq20$: $$\begin{align} 2! &= 2^\color{red}{1} &S_e=1\\ 3! &= ...
12
votes
1answer
1k views

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

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for ...
0
votes
0answers
78 views

conjectured generalization of euler's formula.

Given the elliptic modulus $k$ ,such that the complementary modulus is defined by $k'\equiv \sqrt{1-k^2}$,the jacobi amplitude $\phi\equiv am(u|k)$ and $K(k)$,is the complete elliptic integral of the ...
2
votes
1answer
69 views

A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal ...
0
votes
0answers
15 views

Endowing module with algebra

Given a module $M$ over a ring $R$, is it possible to endow $M$ with an operation $M^{2} \to M$ that turns $M$ to an algebra that is not the trivial $m n \equiv \mathbf{0}$ identically? So far I have ...
3
votes
1answer
72 views

Vladimir Blinovsky's Union-Closed Sets Conjecture Proof

Recently, Vladimir Blinovsky published an article (http://arxiv.org/pdf/1507.01270v6.pdf) claiming that he proved the union-closed sets conjecture ...
1
vote
1answer
15 views

$f$ has to be open in order for continuous bijection to be a homeomorphism?

Claim : Let $f: X\to Y$ be a continuous bijection, then $f$ is a homeomorphism if $f$ is open Is the claim true? I thought $f$ was a homemorphism if $f^{-1}$ is a continuous function
1
vote
0answers
30 views

The Divisors of $s(2s+1)$ and Primes $n$, $4n+1$, and $6n+1$

This question is somewhat related to this one. Most of this is by way of a computer search: claim: If $s$ is any positive integer I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be the divisor ...
3
votes
0answers
280 views

The Divisors of $s(2s+1)$ and Primes $2n+1$ and $3n+1$ part 1

I want to check my math (and proof) on the following claim. The claim is by way of a computer search and a "hunch". claim: If $s$ is a prime number I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be ...
2
votes
3answers
103 views

Conjecture about linear diophantine equations

I've been dabbling with linear Diophantine equations and came across a rather interesting pattern that I would like to conjecture as true but I have no idea how about to come up with a proof. Let ...
1
vote
1answer
43 views

Prime conjecture containing primorial

Help me find the exact conjecture statement. What I roughly remember is that it stated that the difference between primorial $n\#$ (product of first $n$ primes) and "some" larger number than the ...
11
votes
2answers
322 views

A conjecture about traces of projections

Let $M_n$ denote the space of all $n\times n$ complex matrices. Define $\tau:M_n\rightarrow \mathbb{C}$ by $$\tau(X)=\frac{1}{n}\sum_{i=1}^n x_{ii},$$ where of course $X=[x_{ij}]\in M_n$. Recall that ...
15
votes
0answers
466 views

a conjectured continued fraction for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for ...
35
votes
0answers
1k views

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
7
votes
1answer
176 views

Uniqueness or non uniqueness of a pair of natural numbers

Let $1<m<n$ be two natural numbers. Let us call $(m,n)$ a math.se pair if the prime factors of $m$ are the same as those of $n$ and the prime factors of $m+1$ are the same as those of $n+1$, ...
0
votes
2answers
42 views

If $b$ is an odd composite number and $\dfrac{b^2 - 1}{\sigma(b^2) - b^2} = q$ is a prime number, what happens when $q = 2^{r + 1} - 1$?

(Note: An improved version of this question has been cross-posted to MO.) Let $\sigma(X)$ be the sum of the divisors of $X$. For example, $\sigma(2) = 1 + 2 = 3$, and $\sigma(4) = 1 + 2 + 4 = 7$. ...
3
votes
0answers
72 views

How close are we to knowing the rate of convergence to $0$ of $\prod_{p\le x}(1-1/p)^{-1}-e^\gamma\log x $?

This is a question related to an earlier one of mine, which I may answer myself eventually, as I have learnt more about the topic. Despite what one can read on the MathWorld page about Mertens' third ...
0
votes
0answers
143 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 $1$. ...
11
votes
0answers
313 views

Are all totient values 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 ...
0
votes
1answer
29 views

Is the following inequality involving the sum-of-divisors and Euler totient functions true?

First Question Is the following inequality involving the sum-of-divisors $\sigma$ and Euler totient $\phi$ functions true? $$\frac{\sigma(N)}{N} \leq \frac{N}{\phi(N)}$$ Second Question When ...
3
votes
0answers
64 views

$(\sum_i a_i^2)(\sum_i b_i^2)+(\sum_ia_i b_i)^2\geq \sqrt{(\sum_i a_i^4)(\sum_i b_i^4)}+\sum_ia_i^2b_i^2$?

I have no idea about how to prove (or disprove) the following 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\geq ...
4
votes
0answers
40 views

If $|G_1|=|G_2|<\infty$ and $|G_1'|<|G_2'|$, then $|Z(G_1)|\geq |Z(G_2)|$? where $G'$ is the commutator subgroup of $G$.

We know that $G'$ characterization how ``abelian'' of a group because we have a theorem: if $G'=\{e\}$, then $G$ is abelian. I have a conjecture. If there are two finite groups $G_1$ and $G_2$, ...
1
vote
0answers
48 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 a positive integer $M ...
4
votes
0answers
26 views

Are there infinitely many primes in any sequence determined by a $k$ that is not a Sierpinski number?

Consider the sequence of numbers ranging over $n$ of the form $k\cdot a^n + 1$ for a fixed, odd natural number $k$. The number $k$ is considered a Sierpinski number if the sequence determined by that ...
-3
votes
1answer
473 views

$k$-tuple conjecture.

This conjecture is false. See this post Time is running out Suggest notation for Steps 1 and 2. Earn the bonus. For each $k\in\mathbb{Z^{+}}$. Step 1: Create a list $(1,1,1,1...,1)$ of length ...
0
votes
0answers
29 views

An approximate relation between the first and the second zeta zero.

Can you improve on this apparent relation between the first and the second Riemann zeta zero: $$\Im\left(\frac{n \cdot \rho _1}{\pi }\right) \approx \Re\left(\left(1-\frac{1}{(n+5)^{\frac{1}{\rho ...
2
votes
1answer
82 views

How do you refute these conjectures that seem imply contradictory statements?

I've formulated two conjectures that seems to imply a strong result when are combined with well known equivalences of the Riemann hypothesis, and I would like to know how get a disproof of such ...
3
votes
2answers
51 views

I think I've found all roots to $f_k(x)=\sum_{j=1}^k x^j-x^{-j}$ for any $k$ - how to prove it?

Conjecture: The set of unique roots of $$f_k(x)=\sum_{j=1}^k x^j-x^{-j} \;,\;\; x \not=0$$ is given by $e^{i \pi \phi_k}$, where $$\frac{1}{2}\phi_k=\{0, \frac{1}{2}, ...
2
votes
0answers
172 views

A Conjecture Sharper than Cramér's and Firoozbakht's

Notation: $\lfloor\cdot\rfloor$ is floor function; and $\pi(x)$ is the prime-counting function up to $x$. $g_k := p_{k+1} - p_k$ . OEIS sequence A267549 is "Primes prime(k) such that floor( ...
1
vote
0answers
24 views

Can we capture the definition to be a Mersenne prime in an identity only involving the arithmetic function $S(n)=\sum_{k=1}^n\text{nmod k}$?

Let $S(n)=\sum_{k=1}^n\text{nmod k}$ the sum of remainder function, denoting $\sigma(n)$ as the sum of divisors function, it is know that for each $n>1$ $$S(n)-S(n-1)=2n-1-\sigma(n),$$ is as ...
84
votes
24answers
2k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
1
vote
0answers
37 views

If $\gcd(Z,\sigma(Z))=1$ and $1<N=Z\sigma(Z)$, 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 ...
2
votes
0answers
46 views

Does the Descartes number $D = 198585576189$ have a friend?

Let $\sigma(X)$ be the sum of the divisors of $X$, and denote the abundancy index $\sigma(X)/X$ by $I(X)$. If the equation $I(X) = r/s$ has no solution $X \in \mathbb{N}$, then $r/s$ is said to be an ...