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

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4
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1answer
126 views
+50

Is the Fermat primality test secure enough for very big numbers?

The random variable $X_m$ is the number of trials before $n\notin\mathbb P\wedge n|2^{n-1}-1$ where $n$ is an odd random integer $2^{m-1} < n < 2^m$. Computer simulations makes me believe ...
1
vote
2answers
46 views

Closed form and limit of the integral of a rational function

While trying to answer this question, I wondered whether there could be a way to: (A) Find the closed form of the generalization of integrals $I$ and $J$, that is $$I_n=\int_{-\infty}^{+\infty}\frac{...
5
votes
1answer
94 views

There are at least two solutions such that $2p_n=p_a+p_b$ ($p$ being prime)

I've stumbled across this playing around and summing primes at random during a boring lecture. Is this a known conjecture? Can it be proven? My conjecture: There exists at least one non trivial ...
0
votes
0answers
53 views

Confusion about Saibians article about primes

Here : https://sites.google.com/site/largenumbers/home/1-5/2 Saibian claims that it was proven that infinite many twin primes exist. Did I miss something ? Isn't the twin-prime-conjecture open ?...
21
votes
6answers
3k 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 ...
2
votes
0answers
26 views

Seeking proof to a Hyperbolic polygon conjecture

In the course of writing a(n Honours) thesis, I'm searching for a proof to a conjecture that appears very likely to be true. Many results will rely upon it. My own attempts to prove it have been ...
0
votes
0answers
32 views

Conflicting conjectures [duplicate]

I feel like when two conjectures are inconsistent with one another, it's a clear sign of our misunderstanding of deeper mathematics. I was wondering if anyone knew of a comprehensive list of ...
0
votes
0answers
26 views

Legendre's Conjecture Theme (Part II)

This is a continuation of this question. My main question is that, in the previous question we were mainly concerned about the sign of, $$f_{2}(n)=\pi\left((n+1)^2\right)+\pi\left(n^2\right)-2\pi\left(...
2
votes
0answers
64 views

Legendre's Conjecture Theme (Part I)

Main Question Recently I have been thinking about the Legendre's Conjecture. I noticed that a proof of the conjecture can be obtained if we can prove any one of the following, Conjecture 1. For ...
6
votes
0answers
222 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 $ord_n(k)$ ...
1
vote
0answers
61 views

Artin's Conjecture on Primitive Roots and the Divisors of $s(2s+1)$

Artin's conjecture on primitive roots state that given and integer $k$ that is not a perfect square or equal to $-1$ is a primitive root modulo infinitely many primes. claim: If $s$ is any ...
0
votes
1answer
80 views

Collatz Conjecture? [closed]

I am a lover of Math, all kinds really it is a bit of a puzzle to me I'm always trying to learn something new or a new "puzzle" to try an solve for myself even though many times i just reach the same ...
2
votes
1answer
47 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 ...
3
votes
1answer
217 views

Conjecture about Rabin-Miller pseudo prime test

I tested the Rabin-Miller pseudo prime algorithm using a single test value and found that the number of false calls depends on the size of the number to test, reducing to a (conjectured) negligible ...
8
votes
0answers
156 views

Open mathematical questions 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? ...
7
votes
2answers
282 views

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 $(3,4),(4,3) \text{ and } (4,4)$...
0
votes
1answer
123 views

A stronger form of Rolle's Theorem in the direction of number of roots of $f'(x)$

Today I read an interesting generalization of the Rolle's Theorem for Polynomials in $E. 28$ of E. J. Barbeau's book on Polynomials. It says that if $a, b$ are two consecutive zeroes of polynomial $P(...
0
votes
0answers
71 views

a continued fraction related to the exponential function $e^x$

Given a natural number $n$,with $|x|\lt1$ define the following conjectured identity $$G(n,x)=\begin{aligned}\cfrac{-n}{1-x-\cfrac{(1+n)(1-x^2)}{1-x^3-\cfrac{x^2(1-x)(1-x^3)}{1-x^5-\cfrac{x^3(1-x^2)(1-...
0
votes
2answers
115 views

A question about the $3n+1$ conjecture

So I know that if you take any even number $n$ that is a power of $2$ like $32 = 2^5,16=2^4$ or $64=2^6$ we will keep dividing by 2 until we reach 1. and so all the steps will be $\frac{n}{2}$ and we ...
0
votes
2answers
73 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 ...
0
votes
0answers
15 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 inverse?...
2
votes
1answer
54 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 ...
3
votes
1answer
214 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
33 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
127 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 $a,b$, ...
83
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
22 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 http://www.ams.org/journals/...
2
votes
0answers
41 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
134 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
57 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. ...
0
votes
0answers
29 views

Examples of Beal's conjecture for higher powers

Does anybody know of any examples of Beal's conjecture for higher powers? I found this web page that has probably the most examples I've come across so far. However, I'd like to find some examples, ...
1
vote
1answer
74 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
51 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
49 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! &= 2^\color{red}...
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 $\displaystyle\cot\left(...
0
votes
0answers
79 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
71 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 E_n=\frac{(2\phi)^n-(-1)^...
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
115 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 (https://en.wikipedia.org/wiki/Union-...
1
vote
1answer
16 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
3
votes
0answers
300 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
109 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 $ax+...
1
vote
1answer
45 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
329 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
475 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 $\displaystyle\tan\left(...
36
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
178 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 $n+1$, e.g....
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 ...