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
1,171
questions
2
votes
1
answer
72
views
Possible connection between binary numbers and $\pi$
Here is the Desmos if you want to follow along: https://www.desmos.com/calculator/b4vtzruupm
In messing around with binary numbers, I created a function $f(x)$ in Desmos that generated a list of ...
- 139
0
votes
0
answers
30
views
Question about the Erdős–Faber–Lovász conjecture
In graph theory, the Erdős–Faber–Lovász conjecture is a problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972. It says:
If k complete graphs, ...
- 193
0
votes
0
answers
28
views
Prove this conjecture: Two lists of vectors are in the same orientation iff this transformation exists
Can you please help prove or disprove my conjecture below?
Definition: Given vector space $V$ of $n$ dimensions, and ordered lists $x, y$ such that $x,y \subset V$; $x,y$ are each linearly independent;...
- 2,161
-2
votes
1
answer
102
views
Collatz conjecture - Why does it end and not go on to infinity? [closed]
I was messing around with the sequences of odd numbers in the Collatz conjecture, and (unsurprisingly) found a pattern. Basically, I was calculating the number of steps an odd number takes to reach an ...
6
votes
1
answer
114
views
Conjecture: Shallow diagonals in Pascal's triangle form polynomials whose roots are all real, distinct, and in $(-2,2)$.
Here are the first few "shallow diagonals" in Pascal's triangle.
We can use these shallow diagonals to make polynomials. Note that the even degree polynomials have an extra $\pm x$ at the ...
- 8,500
0
votes
2
answers
75
views
Conjecture about the minimum of the Gamma function
Problem/Conjecture:
Let the function :
$$f(x)=\frac{((x+x_{\min})!-(x_{\min})!)^{\frac{1}{x}}}{x^{\frac{1}{x^2}}}$$
Where $x_\min$ denotes the minimum abscissa of the Gamma function near by $0.4616$
...
- 3,243
6
votes
1
answer
115
views
Formula for the ratio of Numbers not divisible by the first n primes
I was playing around with prime numbers, and specifically how many prime numbers we can exclude from an interval from being primes. It is easy to see, that after 2, the ratio of numbers is $1-\frac{1}{...
- 81
4
votes
1
answer
63
views
Friendship theorem for bipartite graphs
The friendship theorem states that if every pair of people has exactly one common friend, then one person is friends with everybody. Which is unfortunately not easy to actually proof. I came up with ...
- 193
0
votes
0
answers
68
views
Strengthening of FLT [duplicate]
FLT is that there are no positive nontrivial integer solutions to the Diophantine equation $a^n + b^n = c^n$ for $n>2$. What about the conjecture that there are no nontrivial solutions to the DE $a^...
- 137
0
votes
0
answers
25
views
Conjectures with the strongest numerical evidence later proven to be false [duplicate]
For a number of conjectures in number theory, the truth of some statement has been checked up to extremely large values. E.g. There are no odd perfect numbers below $10^{1500}$
Are there any examples ...
- 1,460
1
vote
1
answer
69
views
Certain primes $p$ which seems to be $\equiv 1\pmod{10}$
$A=\{5,61,181,1741,36721,60901,135721,431521,531481,552301,685621,2834581,3567121,3674761,3696481,4503001,6121501,6811741,9456901,11002741,11524801,12495001,15629641,16068781,18611101,20218441,...
- 13.5k
0
votes
1
answer
28
views
Every bipartite graph is an induced subgraph of a hypercube graph?
UPDATE: THIS IS FALSE.
I came up with this question some day and have been working on it for some time:
Every bipartite graph $G$ is an induced subgraph of a hypercube graph $Q_n$.
Here the ...
- 460
1
vote
1
answer
37
views
A graph where for each pair of vertices, there exist a vértice adjacent to both and a vértice adjacent to neither
Can we construct a simple $G=(V, E) $ so that for any pair $u, v\in V $, there is a vértix $x$ adjacent to both, as well as a vertix $y$ that is not adjacent to either $u, v$. So, $ux, vx \in E$ and $...
- 193
5
votes
1
answer
290
views
Are there an infinity of consecutive primes that have no common digit?
Examples: $2$ & $3$, $3$ & $5$, $5$ & $7$, $59$ & $61$, $99999989$ & $100000007$.
I was inspired by the fact that, in English, all consecutive numbers share a common letter, such ...
- 137
4
votes
1
answer
203
views
Revisiting MSE question 4386812 - Determining whether $\sigma(q^k)/2$ is squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$
Preamble: The present inquiry is an offshoot of this earlier MSE question.
MOTIVATION
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the ...
- 10.7k
0
votes
0
answers
25
views
Some conjectures on additive and subtractive bases
I came up with a few problems about additive bases that might be interesting, would love to see if someone can prove or disprove them.
Let S be a set of whole numbers so that every natural can be ...
- 193
2
votes
0
answers
100
views
Every twin prime average $x \gt 6$ is the sum of two twin prime averages (Code checked up to $x \leq 1,000,000$).
If $p,q$ are a pair of twin primes, then $x = \dfrac{p+ q}{2} = q-1 = p+1$ is their twin prime average.
Conjecture. Every twin prime average $x \gt 6$ is the sum of two smaller twin prime averages, $...
- 19.8k
2
votes
1
answer
102
views
0
votes
0
answers
28
views
Diagonal patterns in a plot related to Artin's Conjecture about primitive roots
This image displays the following information. The pixel in the ($a$-th row, $k$-th column) is
\begin{equation}
\begin{cases}
\text{gray} & \text{if } a \geq p_k \\
\text{white} & \text{if } a ...
- 17
12
votes
1
answer
320
views
Is $5^x+x$ ever prime? [duplicate]
Are there any integers k for which $5^k+k$ is prime? A simple estimate says there should be, but I can't find one.
I was querying ChatGPT to give a novel mathematical conjecture. After 6 unsuccessful ...
- 878
15
votes
1
answer
491
views
Conjecture about difference of Fibonacci numbers and primes
I'm curious to see if this conjecture is true: if $ m > 4 $ is a positive integer not divisible by $ 2 $ or $ 3 $, it's possible to find a positive integer $ n $ such that the difference of the $ 2 ...
- 409
-2
votes
3
answers
135
views
(Almost) found a counterexample to the Beal conjecture [closed]
The Beal Conjecture states that if $A^x + B^y = C^z$, where $A, B, C, x, y, z$ are positive integers and $x$, $y$, $z$ are $≥ 3$, then $A$, $B$, and $C$ have a common prime factor.
There is some ...
- 4,913
3
votes
0
answers
35
views
Do all Turing-complete systems converge on a single universal function w.r.t. description redundancy?
Let $L$ be any Turing-complete language.
Let $d_L(n)$ be the number of distinct algorithms expressible in $L$ using at most $n$ bits.
I'm not sure how to properly define "distinct algorithm",...
- 5,504
2
votes
1
answer
49
views
Graphs with all vertices having more neighbors than 2nd order neighbors
This is the conjecture that inspired my question:
https://en.wikipedia.org/wiki/Second_neighborhood_problem
For any vertex $v$, let $N(v)$ be the set of vertices adjacent to $v$ and let $N^2(v)$ the ...
- 193
0
votes
0
answers
75
views
What is the function E(x)?
I’m an electrical engineering student getting a minor in math. I recently started my first pure math class and my professor proposed something interesting on the board. I don’t remember exactly what ...
1
vote
1
answer
102
views
If $p^k m^2$ is an odd perfect number with special prime $p$ and $p = k$, then $\sigma(p^k)/2$ is not squarefree.
While researching the topic of odd perfect numbers, we came across the following implication, which we currently do not know how to prove:
CONJECTURE: If $p^k m^2$ is an odd perfect number with ...
- 10.7k
-1
votes
1
answer
43
views
Given the $2^n$ primorial divisors, $D(5\#) = \{ 1, 2, 3, 5, 6, 10, 15, 30\}$, why are there always exactly $2^n$ different divisor indicator vectors?
Define $D(m)$ to be the (ordered) set of divisors of $m \in \Bbb{N}$.
Let $m = p_n\# = 1 \cdot 2 \cdot 3 \cdot 5 \cdots p_n$. Clearly, $|D(m)| = $ the number of divisors of $p_n\# = 2^n$.
Now define ...
- 19.8k
0
votes
0
answers
61
views
How can I determine the reason for the mismatch between the value of this infinite continued fraction and the expected result?
How can I determine the reason for the mismatch between the value of this infinite continued fraction and the expected result?
$$6.9431070487= \lim\limits_{n\rightarrow\infty}7- \frac{7}{129- \dfrac{...
user1135362
-1
votes
1
answer
175
views
Conjecture about the representation of a constant $C=0.6516...$
It's a follow up of my previous question How to find the constant $C$ such that $f(x)\geq Cx$ :
We start with :
$$\left|\exp\left(1-\prod_{k=1}^{1000}\left(1-\frac{1}{2^{k}(k+1)}\right)\right)-\sqrt{2}...
- 3,243
0
votes
0
answers
65
views
proving an equation relating two modular arithmetic inequalities
I made an observation, (which is significant in another problem) but I don't know how to go about it.$\ $
below $a\%b$ stands for $a\mod b\ $. If we call
$m$ the smallest positive integer such that $...
- 175
3
votes
0
answers
192
views
Why is it not known that $\sum_{n>0}\frac{(-1)^nn}{p_n}$ converges?
No one knows if this sum $$\sum_{n>0}\frac{(-1)^nn}{p_n}$$converges (where $p_n$ is the $n$th prime). But what makes proving this so hard? We know that $$\bigg|\sum_{n>0}\frac{(-1)^n}{p_n}\bigg|&...
- 3,574
40
votes
1
answer
674
views
Regular polygon of radius $1$ with diagonals: mysterious ring of radius $1/e$?
I was playing with a geogebra applet that shows regular $n$-gons of radius $1$ with their diagonals. For example, here is the $12$-gon with its diagonals:
For any value of $n$, when I shrink the ...
- 8,500
2
votes
1
answer
56
views
Proving a variation of Lemoine's Conjecture by assuming the strong Goldbach Conjecture
In 2013, when I was just a totally newbie recreational mathematician, I read about Levy's conjecture (i.e., Lemoine's conjecture, stating that all odd integers greater than 5 can be represented as the ...
- 543
3
votes
0
answers
88
views
Filling space with polycube snakes
One of Martin Gardner's "Mathematical Games" columns (Scientific American June 1981, pp24–29; reprinted in The Last Recreations (1997), pp274–283, and The Colossal Book of Mathematics (2001) ...
- 553
4
votes
0
answers
136
views
Is $650$ the only solution not fitting in the family?
Inspired by this question
The linked question conjectures that $\frac{\sigma(n)}{n+1}$ (where $\sigma(n)$ denotes the divisor-sum function) is not an integer for any squarefree composite number.
If we ...
- 80.6k
0
votes
1
answer
55
views
Conjecture: Prove that, $\forall (m,n,k)\in\Bbb Z_{>0}^{3}$ there exists $x\in \Bbb Z_{>0}$, such that: $m^x \equiv x^n\pmod k$
Conjecture: Prove that, $\forall (m,n,k)\in\Bbb Z_{>0}^{3}$ there exists $x\in \Bbb Z_{>0}$, such that:
$$m^x \equiv x^n\pmod k$$
The conjecture seems correct.
Also this conjecture is possible ...
- 1,774
29
votes
1
answer
907
views
If $n$ is square-free and $n+1 \mid \sigma(n)$, is $n$ a prime?
As the title says, is $n$ being square-free together with $n+1 \mid \sigma(n)$ sufficient to show $n$ is prime?
It is well-known that if $\varphi(n)\mid n-1$ and $n+1 \mid \sigma(n)$, then $n$ is a ...
- 14.8k
2
votes
0
answers
50
views
Any "reasonable" and "natural" false graph theory conjectures for which the Petersen graph is not a counterexample?
I heard that the Petersen graph is a good test of a graph theory conjecture. As in, if you want to know whether a graph theory conjecture is false, the Petersen graph is likely to be a counterexample. ...
- 16.8k
2
votes
0
answers
95
views
Is this stronger conjecture about near-rep digit prime numbers true?
Inspired by this post, I established an even stronger conjecture:
If we have the number $1\ldots 1$ with $k\ge 3$ digits $1$ in decimal expansion, then we can replace a digit which is neither the ...
- 80.6k
9
votes
2
answers
478
views
Conjecture: There always exist $k\in \Bbb N$, such that $m^k\equiv k\pmod n$, where $m,n\in\Bbb N.$
Conjecture:
Let $m,n\in\Bbb N$. Then there always exists $k\in \Bbb N$, such that $m^k\equiv k\pmod n$ holds.
This question comes from here.
Let $m,n\in\Bbb Z_{>0}$ are fixed numbers, such that $...
- 1,774
2
votes
0
answers
73
views
Coman's Last Conjecture stating that every prime $q \geq 11$ can be written as $3 \cdot (p_1-1) + p_2$, where both $p_1$ and $p_2$ are prime numbers.
Today I was taking a look at Coman's book entitled Conjectures on Primes and Fermat Pseudoprimes, many based on Smarandache function (starting from the end, as I often do) and his last conjecture, the ...
- 543
2
votes
0
answers
34
views
How proof The shape of the two hearts in $\mathcal{D}^b(A_n)$ is the same if the graded undergrphas of the Ext quiver of heart of are equal
According to Yu Qiu 's Ext-quivers of hearts of A-type and the orientation of associahedron,
We know hearts in $\mathcal{D}^b(A_n)$ one by one corresponds to the Ext quivers and are precisely the ...
- 29
1
vote
1
answer
122
views
On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$
Let $N$ be an odd perfect number given in the so-called Eulerian form
$$N = q^k n^2$$
where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
In what follows, let ...
- 10.7k
2
votes
1
answer
63
views
Calculation of generalized Artin's constants
Let $T(p)$ be the period of the decimal expansion of $1/p$, for prime $p$ (e.g. $1/7=0.\overline{142857}\rightarrow T(7)=6$). It is known that
$$T(p)=\frac{p-1}{t}$$
for some integer $t$. Then, Artin'...
- 979
1
vote
1
answer
28
views
Find conjugate function
$A\in R^{m\times n},\ rankA = m,\ f(y) -\text{ Convex function,} \ f:R^m\rightarrow R.$ Let $g(x) = f(Ax)$
Find $g^*(y)$.
Because$x\rightarrow Ax$ - Affine transformation $\implies g(x)$ - is convex ...
- 29
13
votes
7
answers
847
views
Prove or disprove: If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$, then $\lim_{n\to\infty}\sum_{k=1}^n f(k/n)$ does not exist.
I'm trying to prove or disprove the following conjecture:
If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$ then $L=\lim\limits_{n\to\infty}\sum\limits_{k=1}^n f\left(\frac{k}{n}\...
- 8,500
5
votes
1
answer
212
views
How many time do we need to shuffle two decks red and blue to get back to initial colors?
Question
Let's say:
I have two decks of $n$ cards each (a blue and a red decks), I am not interested in card faces, just in colors of card backs;
Each time I shuffle perfectly ($n$ cards in each hand,...
- 480
6
votes
2
answers
185
views
$\sum{x^{(i,n)}}+1$ is irreducible?
A simple calculation using Burnside lemma shows that the number of distinct "necklaces" with $n$ balls of $x$ colors is
$$\frac{F_n(x)}{n} = \frac{\sum_{i=1}^n x^{(i,n)}}{n} = \frac{\sum_{d \...
- 460
1
vote
0
answers
106
views
Is this conjecture well-formed?
For each positive integer $n$, one of these is true, where $a,b,c,...etc.$ are natural numbers:
$$
\begin{array}{l}
3^0\cdot{n}=2^a\\
3^1\cdot{n}=2^a-(3^0\cdot2^b)\\
3^2\cdot{n}=2^a-(3^0\cdot2^b)-(3^1\...
- 197
2
votes
0
answers
112
views
Questions Related to the Twin-Prime Conjecture
For the sieve of Eratosthenes, let $E_k$ be the number of elements left after removing all primes up to $p_k$ and their multiples from the set $\left\{1,2,3,...p_k\#\right\}$ where $p_k\#=\prod\...
- 5,627