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

learn more… | top users | synonyms

1
vote
1answer
26 views

$\forall p\in\mathbb P\exists q,r\in\mathbb P':p^3=2q+r$, $\mathbb P'=$ set of non twin primes

Define $\mathbb P'=\{n\in\mathbb P|n-2,n+2\notin \mathbb P\}$. Conjecture: Given a prime $p>3$, then $\exists q,r\in\mathbb P':p^3=2q+r.$ Tested for the first 10000 primes.
6
votes
1answer
66 views
+50

$\forall p\in\mathbb P\exists q\in\mathbb P\exists n\in \mathbb N: q-p=2^n$?

Conjecture: $\forall p\in\mathbb P\exists q\in\mathbb P\exists n\in \mathbb N: q-p=2^n$ Verified for the 100 first primes.
0
votes
1answer
42 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 $...
1
vote
2answers
115 views

Primes in the binomial transform of $ [1, 1, 2, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, …]$.

This question is related to this sequence A139482. A commentator gives the following formula for $a_m$ $$a_m = {3m^2-9m+10 \above 1.5pt 2}$$ I have that you should consider the sequence $b_n =3n+2$ ...
2
votes
1answer
31 views

If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 - \frac{5}{3q}$?

Let $\sigma=\sigma_{1}$ be the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q$ satisfies $q \equiv k \equiv 1 \pmod 4$), and $k=1$, does ...
8
votes
1answer
442 views

Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $q\equiv m!\pmod p$ for some $2<m<p$?

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
8
votes
1answer
225 views

Are there infinitely many primes $p$ such that $\frac{(p-1)! +1}{p}$ is prime?

Here I have the following conjecture: Let $$S_1(n)= \frac{(n-1)! +1}{n}.$$ Then there exist infinite prime numbers $p$ for which $S_1(p)$ is prime. And I don't know how to prove it. EDIT Let $...
9
votes
1answer
147 views

Does any sum of twin primes, where the sum is greater than 12, also represents the sum of 2 other distinct primes?

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
1answer
53 views

Prime conjecture containing primorial: the difference between the primorial $n\#$ and the smallest prime $p > n\# + 1$ is always a prime

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 ...
8
votes
2answers
299 views

A conjecture about the prime function $p_n$: $p_m \cdot p_n >p_{m \cdot n}$

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

Is this a known conjecture? Given odd primes $p,q$ with $p + q$ sufficiently large, must there exist a different pair $p',q'$ with $p+q = p'+q'$?

Conjecture: There is a natural number $N\in\mathbb N$ such that given odd primes $p,q$ with $p+q>N$ there are primes $p',q'$ where $p' \notin \{p,q\}$ such that $p+q=p'+q'$. Is this known?
2
votes
1answer
80 views

Does this category have a name? (Relations as objects and relation between relations as morphisms)

Given two relations $R\subseteq A\times B$ and $R'\subseteq A'\times B'$. Is it known/used that every relation $r\subseteq R\times R'$ can be characterized by two relations $\alpha\subseteq A\times A'$...
1
vote
1answer
65 views

For every prime $p > 3$ that is $3$ mod $4$, does $q+1 \mid p-q$ for some other prime $q$?

Yet another random conjecture about primes: Given a prime $p>3$ of the form $4n+3$. Then there exist a prime $q<p$ such that $q+1\mid p-q$. Verified for all $p<100000$.
4
votes
0answers
61 views

Does every power of two arise as the difference of two primes?

Conjecture: For each $n\in\mathbb N$ there are primes $q<p$ with $p-q=2^n$. Verified for $n\leq 26$: ...
2
votes
0answers
138 views

A conjecture about primes: if $a,b$ are coprime and not both odd, is $A(a,b,m)$ finite for some $m$?

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 $a,b$, ...
25
votes
4answers
612 views

Does this conjecture about prime numbers exist? If $n$ is a prime, then there is exist at least one prime between $n^2$ and $n^2+n$.

I made an observation on prime numbers, want to check if any conjecture already exist or not? I am a computer programmer by profession and I am interested in number theory. As like many others I am ...
4
votes
3answers
234 views

Does the sequence $q(n)=3n+1+\frac{1-(-1)^n}{2}$ generate all the prime numbers?

Define $$q(n)=3n+1+\frac{1-(-1)^n}{2} \quad, \quad n\in \mathbb N.$$ $$1,5,7,11,13,17,19,23,25,29,31,35,\dots$$ It seems like this formula gives all primes $>3$ (although not just primes of ...
0
votes
0answers
26 views

Combining a working hypothesis for odd perfect numbers with an inequality for logarithms

Euler's theorem for odd perfect numbers states that if there exists and odd perfect number, that is an odd positive integer $n$ satisfying $\sigma(n)=2n$, where $\sigma(m)=\sum_{d\mid m}d$ denotes the ...
14
votes
3answers
1k views

A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$

If $\;p=m+n$ where $p\in\mathbb P$, then $m,n$ are coprime, of course. But what about the converse? Conjecture: $p$ is prime if $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$ ...
9
votes
3answers
1k views

Erdős-Straus conjecture

I'm reading a lot about the Erdős-Straus Conjecture (ESC), a conjecture that states that for every natural number $p \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following ...
1
vote
0answers
28 views

Graph Laplacian Rank-One update

Can anyone help me prove/disprove this conjecture? Let $G$ be an undirected nonnegative weighted connected graph with $n$ nodes and Laplacian matrix $L$. Also, let $0=\lambda_1<\lambda_2\leq \...
1
vote
1answer
57 views

Is $\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}$ already known?

Is it already shown or at least conjectured that $$\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}?$$ You have any names and years who proved or conjectured it?
1
vote
0answers
22 views

Can this heuristic about Sorli's conjecture and odd perfect numbers be made rigorous?

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. That is, we have $q \equiv k \equiv 1 \pmod 4$. Sorli (page 89) conjectured that $k=1$ always holds. Suppose we rewrite $N$ as $$N = ...
1
vote
1answer
84 views

Divisors of $s(2s+1)$ and the primes $s^2+1$

I need help proving that $s^2+1$ is prime in the following claim. claim: If $s$ is any positive integer I write $f_s =s(2s+1)$. I will need the divisor counting function $\tau$. Suppose that $...
1
vote
0answers
64 views

What are the mathematical consequences if $10$ is proved to be solitary?

Let $\sigma(x) = \sigma_{1}(x)$ denote the sum of the divisors of $x$, and let $$I(x) = \dfrac{\sigma(x)}{x}$$ be the abundancy index of $x$. For example, $$\sigma(10) = 1 + 2 + 5 + 10 = 18$$ so that ...
2
votes
0answers
152 views

Kolmogorov 0-1 Law Converse?

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 $1$. ...
3
votes
0answers
47 views

On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers

(Note: This question has been cross-posted from MO.) Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$. A number $M$ is called almost perfect if $\sigma(M) = 2M ...
1
vote
1answer
73 views

Conjecture concerning modular arithmetic

Below $0\notin\mathbb N$. I want a proof or a counter-example of the following (corrected) conjecture: Suppose $p$ is the smallest prime dividing $n\in\mathbb N$ and suppose $kn+ap=m!$, where $...
4
votes
1answer
133 views

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
50 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
100 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 ...
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
28 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
33 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
28 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
72 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
223 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 $ord_n(k)$ ...
0
votes
1answer
93 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
157 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? ...
1
vote
1answer
126 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
73 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
116 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
76 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 ...
4
votes
1answer
219 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
34 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 ...
84
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 ...