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

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3
votes
1answer
99 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
votes
1answer
99 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
vote
1answer
65 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, $$\begin{aligned}M(q)=\cfrac{1-q^3}{1-q^...
2
votes
1answer
36 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 ...
0
votes
2answers
39 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 ...
1
vote
0answers
44 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 ...
7
votes
4answers
3k views

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
votes
1answer
59 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
votes
0answers
121 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 ...
0
votes
0answers
52 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 \mathbb{...
3
votes
0answers
85 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
votes
1answer
42 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 Pell-...
4
votes
0answers
59 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?
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
1answer
36 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 ...
15
votes
0answers
484 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(...
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-...
3
votes
2answers
234 views

I conjecture that there are infinitely many linear relations $p_n + p_{n + 3} = 2 p_{n + 2}$ in the sequence of primes!

Let $p_1 < p_2 < p_3 < \ldots < p_n < \ldots$ be the sequence of primes (with $p_1 := 2$ as usual). Conjecture: There are infinitely many positive integers $n$ for which \begin{...
8
votes
2answers
162 views

the ratio of jacobi theta functions and a new conjectured q-continued fraction

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define $$\begin{aligned}H(q)=\cfrac{2(1+q^2)}{1-q+\cfrac{(1+q)(1+q^3)}{1-q^3+\cfrac{2q^2(1+q^4)}{1-q^5+\cfrac{q^3(1+q)(1+q^5)}{1-q^7+\cfrac{q^...
11
votes
2answers
252 views

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

There are few known closed form for values of the dilogarithm at specific points. Sometimes only the real part or only the imaginary part of the value is known, or a relation between several different ...
7
votes
2answers
257 views

a conjecture of certain q-continued fractions

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define, $$\begin{aligned}F(q)=\cfrac{1-q^2}{1-q^3+\cfrac{q^3(1-q)(1-q^5)}{1-q^9+\cfrac{q^6(1-q^4)(1-q^8)}{1-q^{15}+\cfrac{q^9(1-q^7)(1-q^{11})}...
3
votes
0answers
101 views

conjectured identity of the product of two theta functions

Looking into the discussion in this post,I was naturally led to consider the following general identity Given the two jacobi theta functions,$$\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}$$ and $...
4
votes
0answers
114 views

A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
1
vote
1answer
33 views

A number-theory question on the deficiency function $2x - \sigma(x)$

Let $\sigma(x)$ be the sum of the divisors of a (positive) integer $x$. (For example, $\sigma(2) = 1 + 2 = 3$.) Define the deficiency function $D(x)$ to be the number $$D(x) = 2x - \sigma(x).$$ Let ...
14
votes
2answers
236 views

Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$

I numerically discovered the following conjecture: $$\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)\stackrel{\color{gray}?}=\frac{7\pi^2}{48}-\frac{\arctan^22}3-\frac{\arctan^23}6-\frac18\ln^2\!...
3
votes
1answer
226 views

Understanding Erdös Discrepancy Conjecture $\left| \sum_{i=1}^k x_{i\cdot d} \right| > C$ for random series of $-1$ and $1$

In today's newspaper NRC from The Netherlands a two-page article was dedicated to computer-aided mathematical proofs (sorry, paid link and in Dutch). Interesting in itself, but as an example it used ...
25
votes
4answers
613 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 ...
1
vote
0answers
38 views

A series summation

How should I calculate the following expression? \begin{equation} T\sum_{n=-\infty}^\infty\frac{1}{\sqrt{k^2+(2\pi nT)^2}}-\int_{-\infty}^\infty\frac{d\omega}{2\pi}\frac{1}{\sqrt{k^2+\omega^2}} \end{...
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 ...
3
votes
0answers
55 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
1
vote
1answer
58 views

For every natural integer $N>3$ there are at least two distinct prime numbers $p$ and $q$ such that $\dfrac{p+q}{2}=N$ and $N-p=q-N$, $(p<q)$.

I'm not sure but this problem may be similar or related to Goldbach conjecture? Any proof/disproof, insight and opinion is appreciated, thanks.
2
votes
0answers
78 views

A false conjecture by de Polignac

(This question would be similar to my other on Goldbach's conjecture so I'll change the "rules") In 1848 de Polignac claimed that "every odd integer is the sum of a prime $p$ and a power of $2$". For ...
0
votes
0answers
53 views

Conjecture on different type of triangle in a complete graph?

How many different triangles are there in $K_5$? The Answer is 35.(The Moscow Mathematics Puzzle) Then I asked what about $K_6$, $K_7$ and so on ...? With my intuition I arrived at this ...
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)^...
31
votes
2answers
644 views

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln n+\frac1{...
24
votes
2answers
962 views

A false conjecture by Goldbach

In 1752 Goldbach send this conjecture to Euler: "Every odd integer can be written in the form $p+2a^2$ where $p$ is a prime or $1$ and $a$ is a natural number (can be even 0)." This conjecture turned ...
3
votes
1answer
57 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 ...
3
votes
1answer
92 views

an interesting q-series and a certain continued-fraction

My aim is to find a rigorous proof of the following conjectured identity.Given $$1+q+q^2-q^4-q^5+q^7+q^8-q^{10}-q^{11}+\ddots=\cfrac{1}{1-q+\cfrac{(q^3)}{1-q^3+\cfrac{q^2(1+q)(1+q^3)}{1-q^5+\cfrac{q^...
2
votes
1answer
58 views

Infinitely many correct solutions to equation? [duplicate]

I conjecture that there are infinitely many correct solutions to this equation: (Where we are assuming $a,b \in \Bbb{N}$) $$a!+1=b^2$$ I chose to list the first three solutions below: $4!+1=5^2$ $5!+...
1
vote
2answers
168 views

a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
5
votes
1answer
198 views

conjectured new generating function of fibonacci numbers

I conjecture a new generating function for the fibonacci numbers $F_{n}$. Given,the following conjectured q-continued fraction $$\chi(q)=\cfrac{1}{1+q-\cfrac{(1+q^2)}{1+q^3+\cfrac{q^2(1-q)(1-q^3)}{1+...
2
votes
1answer
76 views

connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff $$\prod_{n=...
4
votes
1answer
156 views

conjecture about primes and a certain q-series.

Using wolfram Mathematica ,I observed an interesting and surprising property concerning prime numbers and q-series which I could not prove.Yet there is strong evidence supporting it. I would be happy ...
1
vote
1answer
107 views

Can the sum of 3 unique primes be expressed as the sum of 2 primes?

Let's consider the example, $$ 3 + 11 + 19 = 33 \\ 2 + 31 = 33 $$ we can see that there are cases where the sum of 3 primes be expressed as the sum of 2 primes. However, I couldn't find a case ...
1
vote
0answers
75 views

If $(a_n)$ is increasing and $a_n^{1/c^n}\to\infty$, then $\sum\frac1{a_n}$ is irrational?

I $\DeclareMathOperator\lcm{lcm}$am trying to generalise the result from this question: If $(a_n)$ is increasing and $\lim_{n \to \infty} a_n^{1/2^n} = \infty$, show that $\sum_{k=1}^{\infty} 1/{a_n}$ ...
0
votes
1answer
69 views

Improved Betrand's postulate

I want to show that $2p_{n-2} \geq p_{n}-1$... Bertand's postulate shows us that $4p_{n-2}\geq p_{n}$ but can we improve on this? any ideas?
3
votes
1answer
56 views

Sum involving the number of zeros of $k$

I'm currently interested in sums involving digit-functions. Especially, I'd like to calculate the following sum: $$ s=\sum_{k=1}^{\infty}{\frac{a_0(k)}{k\left(k+1\right)}} $$ Where $a_0(k)$ is the ...
12
votes
0answers
169 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ &+720\ln^32\cdot\ln3+360\ln2\cdot\ln^...
1
vote
0answers
35 views

The asymptotic behaviour of triples $n!+q^{n!}=c$, where $q$ is the first prime greater than $n$, and abc conjecture

For a large positive integer $n$, let $q=q(n)$ (below we denote this $q=q(n)$ by $q_{N}$ because we assumed that is the $N$-th prime number) the first prime number which is (strictly) greater that $n$....
1
vote
0answers
65 views

Beal Conjecture and ($\bmod 3$) operation [closed]

When we apply a ($\bmod 3$) operation on the $A^x +B^y =C^z$ we will see some strange results. For e.g.: Let $A=6m+1$ & $B=6n+1$. Since $A$ & $B$ are odd numbers, $C$ will have to be even. ...