Questions tagged [conjectures]

For questions related to conjectures which are suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found

Filter by
Sorted by
Tagged with
3 votes
2 answers
72 views

A function asymptotical equivalent with the prime counting function?

Let $p_n$ be the $n$-th prime number and $Q_a(N)$ be the number of primes of the form $p_n^2+a$ where $1\leq n\leq N$ and $a$ is positive and even. For some $a$ like $26,56$ it seems that no solutions ...
user avatar
  • 13.3k
2 votes
1 answer
76 views

Conjecture about areas of circular segment and polygon with equal perimeter sharing a side

I was playing around with shapes and have formed a conjecture. Length of the red circular arc $=$ total length of the $n$ green line segments Conjecture: $$\sup{\left(\frac{\text{Area}_1}{\text{Area}...
user avatar
  • 1,192
1 vote
0 answers
66 views

fundamental solution of Pell’s equation under special conditions

I make the following conjecture from an answer. conjecture Suppose $N$ could be expressed as $$ N= p^2\pm q,\ \ \ \ (p,q \in \mathbb{N},\ 1 \lt q \lt p,\ q\mid 2p) $$ In this case, the fundamental ...
user avatar
  • 408
4 votes
0 answers
226 views

Why does $e/\pi$ (or $1/2$) arise in this limit?

(Related to this prior post.) Let $f(n)$ for $n\in\mathbb N$ be a function that increases the prime index of each prime factor of $n$ (with multiplicity) by $1$. e.g. $f(20)=f(2^2\cdot 5)=f(p_1^2\cdot ...
user avatar
  • 4,941
4 votes
1 answer
61 views

I have a conjecture regarding vertex coloring

Let $G = (V, E)$ be a graph with chromatic number $n+1$, and let there be some vertex $v* \in V$, so that deleting this vertex results in a graph $G/{v*}$ that has chromatic number n. Now, assign to $...
user avatar
1 vote
0 answers
20 views

Is there anything known about the tree width of the direct sum of two graphs?

So, if you have two graphs (both with a designated edge), you could take the direct sum of those two graphs, where you glue those two graphs to each other along the designated edge. I think different ...
user avatar
2 votes
1 answer
53 views

I conjecture that the circumference of a graph is larger than its average degree

Let $G = (V,E)$ be a simple loop-less graph. The circumference refers to the length of the largest cycle of $G$. The average degree is $avg(G) = |E|*2 / |V|$. I conjecture that if the avg(G) > 2, ...
user avatar
0 votes
1 answer
58 views

Asymmetric graphs where all adjacent vertices have different degree

I conjecture the following : If $G$ is a graph so that every pair of adjacent vertices have different degree, then $G$ is not asymmetric graph. I will remind that a graph is a asymmetric if there are ...
user avatar
7 votes
1 answer
118 views

Proof for strict inequality $\pi(ab) > \pi(a)\pi(b)$?

I asked about the very similar $\pi(ab)\geq \pi(a)\pi(b)$ a while ago, and this is indeed a proven result for $a,b\geq \sqrt{53}$. Empirically, the stricter $\pi(ab)>\pi(a)\pi(b)$ looks true so ...
user avatar
  • 4,941
0 votes
0 answers
30 views

What can we say about the colourings of some subset of vertices of a graph

I will clarify the question by defining some straightforward concepts. Let G = (V, E) be a simple graph with chromatic number $k$. We call a subset $V'\subset V$ fully chromatic if every proper k-...
user avatar
1 vote
0 answers
113 views

Conjecture on ordering the first $p^2$ naturals by prime factor count

Let $\text{bump}(n)$ for $n\in\mathbb N$ be a function that increases the prime index of each prime factor of $n$ (with multiplicity) by $1$. I'll also use the notation $\text{bump}^k(n)$ to signify $\...
user avatar
  • 4,941
3 votes
0 answers
133 views

Conjecture: there are more than $\pi(p)$ primes between consecutive prime squares

For any $p_i$ (being the $i$th prime), it seems empirically certain that $$\pi(p_{i+1}^2)-\pi(p_i^2)>i.$$ Equivalently: there are more than $\pi(p)$ primes between any $p^2$ and the next higher ...
user avatar
  • 4,941
0 votes
0 answers
21 views

Polignac's conjecture and Hardy-Littlewood asymptotic density confusion

I found the Polignac's conjecture here Polignac's conjecture. The conjecture said, if $\pi_n(\mathcal X)$ denotes the number of prime of gaps $n$ less than or equal to $\mathcal X$, then $$\pi_n(\...
user avatar
3 votes
2 answers
85 views

For any composition sequence $s$ of maps $h(X)=X/2, \ f(X)=(3X + 1)/2$, there exists an integer $X$ such that its Collatz sequence contains $s$

Let $h(X) = X/2$, and $f(X) = (3X + 1)/2$. Then clearly every iteration $g^i(X), X \in \Bbb{Z}$ the Collatz mapping $$g(X) = \begin{cases} X/2, \ X=0\pmod 2\\ \dfrac{3X + 1}{2}, \ X = 1\pmod 2 \end{...
user avatar
1 vote
1 answer
50 views

What is the distribution of $ax^2+bx+c$ as $\{a,b,c,x\}$ vary over $\mathbb Z$? (re: quadratic primes)

I realize this may be one of those questions where it's ill-posed and can't be answered without further constraints. But if not, what's the distribution like? In particular, is every integer equally ...
user avatar
  • 4,941
0 votes
1 answer
48 views

How much Base Proof is required?

If I were looking to use the Theory of Strong Induction, to prove the statement that: The Set of All Even Numbers is equal to the Set of All Odd Numbers multiplied by $2^n$, where $n \geq 1$ $$ \...
user avatar
-2 votes
1 answer
152 views

Is the following infinite product of fractions of linear factors equal to an exponential function or not?

Is the following infinite product: $$ \prod_{\substack{(a,b) \in \mathbb{Z}^2 \\ a > b}} \frac{x+b}{x+a} $$ defined? If so, does it simplify to an exponential function of $x$? The subscript is ...
user avatar
5 votes
0 answers
113 views

Conjecture: between any two consecutive squares, there are integers matching each of $2p, 3p,$ and $4p$; also, more terms with higher degrees

This is a minor twist on Legendre's conjecture, of course. To restate: I submit that for all $n>1$, every interval $\left(n^2,(n+1)^2\right)$ contains at least one integer matching each form $p, 2p,...
user avatar
  • 4,941
0 votes
0 answers
23 views

What is the inverse operator of a weighted integral?

So I've been trying to construct the inverse operation of the conjectured solution. Let's say I have the following differential operation: $$ \frac{d_1 f}{d_1 x} = \frac{f(x+ \epsilon) - f(x)}{\...
user avatar
3 votes
1 answer
114 views

If $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1) n^2$, does it follow that $q n^2$ is perfect?

Let $q$ be an (odd) prime, and let $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. A number $N$ is said to be perfect if $\sigma(N)=2N$. ...
user avatar
-1 votes
1 answer
85 views

Observation on Legendre's Conjecture

Legendre's Conjecture states that there is a prime between every two perfect squares. It's almost certainly true, but remains unproven. I conjecture that not only is that true, but for every $a\in\...
user avatar
  • 4,941
3 votes
0 answers
119 views

Primes being a square sum of an odd number and a less number

For natural numbers $a,b$, let $a$ be odd and let $f(a)$ be the number of $\,b<a\,$ such that $a^2+b^2$ is prime. In the diagram $f$ is blue, the function $g(n)=\frac{n}{\ln n}$ is red and $h(n)=\...
user avatar
  • 13.3k
0 votes
0 answers
57 views

Is it true that $l_1(q,n) \geq g(k)$, if $q^k n^2$ is an odd perfect number with special prime $q$?

(Note: This post is an offshoot of these earlier questions: (post 1) and (post 2).) Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(...
user avatar
1 vote
1 answer
108 views

On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number

(Note: This post is an offshoot of this earlier MSE question.) Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the ...
user avatar
2 votes
1 answer
183 views

All solutions to $1/a+1/b=1/c$?

Conjecture: Given integers $a>b>c>1$ such that $\gcd(a,b,c)=1$. Then all the positive integer solutions to $$\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$$ is given by: $c$ any number and for each ...
user avatar
  • 13.3k
2 votes
1 answer
112 views

Generalizing the result $|a\times b|=|a||b||\sin \theta|$ to arbitrary dimensions.

The cross product can be generalized to arbitrary dimensions as done below or here. I'm trying to state and prove the general analogue (for arbitrary dimensions) of the equation $$|a\times b| = |a||b||...
user avatar
  • 3,705
0 votes
2 answers
84 views

Real number version of the complex conjugate theorem?

Recently, I have encountered a question where a polynomial and one of its irrational roots $a+b\sqrt{c}$ where $a,b,c\in\mathbb{Q}, \sqrt{c}\in\mathbb{R}$ were given, and I was asked to find all roots....
user avatar
  • 1,644
1 vote
0 answers
90 views

Conjectured closed form for ${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$

With Maple i find this closed form: ${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$=$-{\frac {{\pi}^{2}}{64}}-{\frac { \left( \ln \left( 1+\sqrt {2} ...
user avatar
  • 193
6 votes
5 answers
412 views

Can two circles intersect each other at right angles, such that one circle passes through the center of the other circle?

More precisely, do there exist two intersecting circles such that the tangent lines at the intersection points make an angle of 90 degrees, and one of the circles passes through the center of the ...
user avatar
  • 105
0 votes
2 answers
94 views

Primes of the form $n^2+n+1$

Let $f(n)=n^2+n+1$. While experimenting I found that Given $m,n\in\mathbb N, \: n>1$. If $f(2n)\in\mathbb P$ and $f((3m+1)n)\in\mathbb P$ then $3|n$. It has ben tested for $0\le m <1000$ and $...
user avatar
  • 13.3k
4 votes
2 answers
184 views

Does $\gcd$ of values of $n$-th degree primitive integer polynomial always divide $n!$?

Let's consider the following conjecture: Let $f(x)=a_0+a_1x+a_2x^2+\dots +a_nx^n$ be a primitive non-zero polynomial with integer coefficients and let $g=\gcd(\{f(k), k \in \mathbb{Z}\})$, then $g$ ...
user avatar
  • 13.2k
2 votes
1 answer
127 views

Irreducible polynomials in $\mathbb Z[X]$ with no primes in the image.

It seems intuitive that if an irreducible polynomial $q\in\mathbb Z[X]$ has no primes in it's image, i.e. $n\in\mathbb Z\implies q(n)\notin \mathbb P$, then there is a non unit integer $m\in\mathbb Z$ ...
user avatar
  • 13.3k
5 votes
2 answers
135 views

A conjecture about non-nilpotent groups

A finite group, $G$, is nilpotent if its upper central series terminates (at $i \in \mathbb{N})$ with $Z^i(G)=G$, where $Z^i(G)$ is it's $i$-th center which can be described as $\{x \in G\mid \forall ...
user avatar
-1 votes
2 answers
198 views

Is $3$ the only prime that is both a Mersenne prime and a Fermat prime?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
user avatar
3 votes
1 answer
153 views

New trigamma identity for $\Psi_1(\frac3{20})+6\,\Psi_1(\frac15)+10\,\Psi_1(\frac25)-\Psi_1(\frac1{20})$

I play with Maple, and I find this relation for the trigamma function: $$\begin{align} \Psi_1\left({\frac{3}{20}}\right)+6\,\Psi_1\left(\frac15\right)+ 10\,\Psi_1\left(\frac25 \right)-\Psi_1\left(\...
user avatar
  • 193
1 vote
1 answer
26 views

Prove conjecture about classification of circle pairings

I need help proving or disproving a conjecture related to circle pairings, which I'm trying to prove for my bachelor final project. I first present some needed terminology and context. A circle ...
user avatar
  • 105
1 vote
0 answers
53 views

Colinear and concurrency configuration in triangle involving arbitrary point

I found out this configuration while fooling around with Geogebra. I tried to solve it on my way back home from work but apparently it's not that easy. Is it a well-known theorem in elementary ...
user avatar
0 votes
1 answer
28 views

Maximum degree for this set of planar graphs where all neighbours of any point lie on a circle centered on that point

Consider an undirected simple planar graph, with these additional restrictions: The plane in which the graph is embedded has a Cartesian coordinate system, and there's an injective function mapping ...
user avatar
0 votes
0 answers
191 views

This is a conjecture, refined through previous responses, about sums of positive integers which define sums of the corresponding cubes.

Revising $4255316$, the conjecture now is: If $$\sum_{i=1}^{i=n}{a_{i}}, i>2$$, divides $$3\sum{x_{i}x_{j}x_{k}}$$, where the product is summed over the$\binom{n}{3}$distinct triplets drawn from ...
user avatar
1 vote
0 answers
131 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part VI

(Note: This question has been cross-posted to MO.) The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\...
user avatar
4 votes
1 answer
88 views

Using a conjecture to solve a conjecture

Is it mathematically correct to use a conjecture to prove another conjecture ? And if the second one is proved, does that mean that we'll only focus on proving the first ? There are many cases that ...
user avatar
  • 477
1 vote
0 answers
134 views

Goldbach-like and Collatz-like conjectures and theorems

I am looking for examples of conjectures and famously hard-to-prove theorems which can be stated in the form: For each natural number $n$, $P(n)$. where $P$ is a predicate of natural numbers that is ...
user avatar
  • 1,638
0 votes
1 answer
125 views

On a possible proof for $p^k < m$, if $p^k m^2$ is an odd perfect number with special prime $p$

Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$. If $\sigma(M)=2M$ and $M$ is odd, then $M$ is called an odd perfect number (hereinafter abbreviated as OPN)...
user avatar
0 votes
2 answers
269 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part IV

(Preamble: This post is an offshoot of this earlier MSE question.) The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\...
user avatar
7 votes
2 answers
109 views

$p = (u + v)^2 + uv$ for primes $p = 5k \pm 1$? [duplicate]

I came up with the following conjecture which I am unable to prove. For all primes $p$ which can be represented as either $5k + 1$ or $5k - 1$ for some positive integer $k$, we can find positive ...
user avatar
1 vote
1 answer
119 views

Potential infinite, fast growing subsequence of twin primes

Probably the most interesting part of this discussion is about twin primes of the form $6x\pm 1$, with $x=4\cdot(5\cdot 7\cdot 11\cdot 13\cdot 17 \cdot 19\cdot 23)$ being a typical example, and the ...
user avatar
12 votes
3 answers
561 views

Does Diophantine equation $1+n+n^2+\dots+n^k=2m^2$ have a solution for $n,k \geq 2$?

When studying properties of perfect numbers (specifically this post), I ran into the Diophantine equation $$ 1+n+n^2+\dots+n^k=2m^2, n\geq 2, k \geq 2. $$ Searching in range $n \leq 10^6$, $k \leq 10^...
user avatar
  • 13.2k
1 vote
0 answers
51 views

Conjectures inspired in the context of Casas-Alvero conjecture, via the logarithmic derivative of derivatives of a polynomial

In the post we assume that $P(x)=a_0+a_{1}x+\ldots+a_{n-1}x^{n-1}+a_{n}x^n$ is a polynomial of degree $1<\deg(P)=d=n$ defined over a field $K$ of characteristic zero. We denote its derivatives as $...
user avatar
0 votes
1 answer
99 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part III

Preamble: This post is an offshoot of this earlier MSE question. The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\...
user avatar
5 votes
1 answer
207 views

The 'inscribed-circumscribed' 6 point circle

Here is a construction I came upon recently: A'B'C' is the contact triangle. X(1) is the Incenter. A'',B'',C'' are the midpoints of the sides of the contact triangle. A''',B''',C''' are the midpoints ...
user avatar
  • 117

1
2 3 4 5
23