Questions tagged [open-problem]
Questions on problems that have yet to be completely solved by current mathematical methods.
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A Number-theoretic Generalization of the Union-closed Sets Conjecture
Denote $\mathbb{N}^*=\{1,2,3,...\}, \dot k = \{k,2k,3k,...\}, \mathbb{P}=\{2,3,5,7,11,..\}$ and write $M\leq \mathbb{N}^*$ to denote that $M$ is lcm-closed, i.e. $a,b\in M\Rightarrow \text{lcm}(a,b)\...
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Open knight's tours on $n \times n \times \cdots \times n \subseteq \mathbb{Z}^k$ boards ($k \in \mathbb{N}-\{0,1\}$)
I would like to know under what conditions there exisits a (possibly open) knight's tour on a generic (hyper)cubic lattice $\{\{0,1,\ldots,n-1\} \times \{0,1,\ldots,n-1\} \times \cdots \times \{0,1,\...
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2
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Research monographs and open problems in universal algebra
I am someone who is very interested in the mathematical subfield of universal algebra. I want to know, what are some significant open problems in universal algebra? I would like a list of such ...
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Has this weak version of Erdős Conjecture on arithmetic progressions been proven, or is it still an open problem?
This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker version of Erdős Conjecture, but I do not know how to prove it.
Erdős conjecture on arithmetic progressions ...
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Five new results on Conway's 99-graph problem [closed]
I realize that this editing won't make the question open (since this is against the guidelines to share results and ask to check them). Meanwhile, I'd like to replace a lot of text with a link so ...
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Is some twin prime average the sum of two twin prime averages, two ways?
Accoring to this question and a linked duplicate, it's been verified empirically up to some number that all twin prime averages greater than six, are the sum of two smaller twin prime averages.
I was ...
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Ruzsa–Szemerédi problem for regular graphs
The Ruzsa–Szemerédi problem asks for the maximum number of edges in a locally linear graph, i. e. a graph in which every edge belongs to a unique triangle (equivalently, any two adjacent vertices have ...
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Theory about NAND gate decompose
Recently, I play a game of Turing completeness where I utilize various gate circuits such as NAND, AND, and NOT to construct a circuit that satisfies the given truth table.
I didn't learn digital ...
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What are some conjectures of your own?
Background: Although this site is most-often used for specific one-off questions, many of the highest scored questions (also on MathOverflow), which gather a lot of attention to the site are about ...
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The set of all sums of 3 primes covers almost all of $2\Bbb{Z}+1$ (result of Helfgott), what can one say about $\Bbb{P} - \Bbb{P} = 2\Bbb{Z}$ problem?
Consider the set $\Bbb{P} = \pm$ the prime numbers and $\Bbb{P}_o$ is similarly $\pm$ the prime numbers other than $2$.
By Helfgott's result on the ternary Goldbach conjecture:
Every odd integer ...
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Asymptotic approximation algorithms for TSP
I have been reading a lot about TSP approximation algorithms recently, and I noticed that most of the algorithms tend to fall under two general categories: some that have a guaranteed approximation ...
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Lambert Pairwise Sums O(n^2), X+Y Sorting
Sorting Pairwise Sums of X+Y with O(n^2) comparisons is a hard problem: http://cs.smith.edu/~orourke/TOPP/P41.html
http://en.wikipedia.org/wiki/X_%2B_Y_sorting
It seems Lambert has concluded that if ...
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Does Han-Kobayashi attain the maximal rate-sum of a discrete memoryless very weak interference channel?
This figure describes an interference channel.
A discrete memoryless interference channel is said to be very weak if:
$$I(U_1;Y_1)\geq I(U_1;Y_2|X_2), ~~\forall ~ (U_1,X_1,X_2) \sim p(u_1,x_1)p(x_2),$...
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Marjorie Senechal (2011): "The question of packing tetrahedra is still unsolved" Is it still unsolved in 2023?
At about 15:49 in her 2011 talk Prof. Marjorie Senechal - "Quasicrystals Gifts to Mathematics":
But Hilbert understood that groups aren't everything and maybe not even the main thing. And ...
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Is equidistant points an open problem?
This post asks whether for any $n$-dimensional (presumably real) normed vector space, you can find $n+1$ equidistant points. They receive two answers saying that it is possible, but neither give much ...
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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, $...
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If $S = A\cdot A\cdot A$ generates $G$ and $f(S) = H$ is a group then $f(A) \leqslant H$ is also a group? Under what conditions is this true?
Let $G, H$ be two abelian groups and $f : G \to H$ a group homomorphism. Suppose that $G = \langle S \rangle$ and that $f(S) = H' \leqslant H$ a subgroup. In that case we say that $S$ represents $H'$...
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If $q^k n^2$ is an odd perfect number, then $n^2 - q^k = 2^r t$ implies that $3 \leq r$ is odd. Therefore?
The topic of odd perfect numbers likely needs no introduction.
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 ...
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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 ...
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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 ...
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Sums of p-th powers of first N positive integers equal a p-th power of an integer
I am looking for $(p, N)$ where $p$, $N$ are integers greater than 1 and
$${\sum_{n=1}^{N}n^{p}}=M^{p}$$
where $M$ is an integer.
$p=2$, $N=24$ leading to $M=70$ is the Cannonball problem, and it was ...
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Summing the kth-nacci sequences over k
I've been playing around with an open problem I found in Peter Winkler's puzzle book. Roughly, it is
Let $C_p(n)$ be the expected length of the longest common subsequence of two random coin flip ...
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The same order type groups
Let $G$ be a finite group and $n$ be a natural number. Set $T(G)=\{g\in G|$ $%
g^{n}=1\}$ and $L_{n}(G)=|T(G)|$. Two finite groups $G_{1}$ and $G_{2}$ are
called of the same order type if and only if $...
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Are there infinitely many composite Euclid numbers of the second kind (Kummer numbers)?
$\displaystyle \prod_{i=1}^n p_i - 1$ is called Euclid number of the second kind (or Kummer number) , where $p_i$ is the i-th prime number.
It is not known whether there are infinitely many prime ...
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Another generalization of open mapping theorem
Let $T: E \to F$ be a linear continuous function between Banach spaces. Let
$$
B_E (x, 1) := \{z\in E \mid |z-x| <1\} \quad \text{and} \quad B_F (y, 1) := \{z\in F \mid |z-y| <1\} \quad \forall ...
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Name of conjecture about correlation of $\lambda(n)$ and $\lambda(n+1)$
I remember reading about a conjecture one night a while ago, but I can’t seem to find anything about it anymore. I have forgotten it’s name, but I believe the conjecture went as follows:
Suppose $\...
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Can we come up with a disjoint union of a subsets of the group $\Bbb{Z}$ such that they do not equal the cosets of a subgroup, yet they form a group?
If this applies to $\Bbb{Z}$ it probably will work for other groups $G$, however, for simplicity and because I'm interested in integers & their primes, let's work with $G = \Bbb{Z}$.
Anyway, we ...
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Doubt about approaching problem from the Kourovka notebook
Problem 20.58 from the Kourovka notebook, particularly the first part, reads
Is it true that for every positive integer $n$ there is a recognizable group that is the $n$-th direct power of a non-...
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Is there any other known relationship between even perfect numbers and odd perfect numbers, apart from their multiplicative forms?
(Note: This was cross-posted from MO, because it was not well-received there. Will delete the MO post in a few.)
Observe that an even perfect number $M = (2^p - 1)\cdot{2^{p - 1}}$ and an odd perfect ...
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Examples of well-known unsolved problems, in subjects don't related to number theory, that have several equivalent formulations
I know statements for some unsolved problems related in some way to number theory, being stated (I refer for each one of these unsolved problems) several equivalent formulations, for example: these ...
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Show that $\dim(Y)<\infty$, application of the Hahn-Banach theorem.
$\textbf{The question is}$
Let $X,Y$ be a Banach space, $T:X\rightarrow Y$ an surjective linear transformation.
If $\exists D\subseteq Y$ compact: $T(B(0,1))\subseteq D$ then $\dim(Y)<\infty$
$\...
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Do three consecutive numbers of form $A^2B^3$ exist?
I (non-mathematician) asked a similar kind of question 5 days ago. Now I revisit the case in a different manner. The powerful numbers may be written in the form $A^2B^3$, where $A$ and $B$ are ...
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Questions about density of $\left\{\ \left( k+ \frac{1}{2} \right)^n \right\}$ in $[0,1]$
Here, $\{ x \}$ denotes the fractional part of $x.$
Are there any known positive integers $k$ for which the set $\left\{\ \left\{
\left( k+ \frac{1}{2} \right)^n \right\}: n\in\mathbb{N}\ \right\} $ ...
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Why is it so difficult to prove $\mathbf{NP} ≠ \mathbf{P}$?
I'm just curious because I saw on Wikipedia a single polynomial time solution to any NP-hard problem would imply there are polynomial time solutions for every single NP problem.
Also I assume there ...
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What makes distributed function computation difficult?
Encoder $k$ informs a decoder of a source $X_k$ at rate $R_k$; $k=1,\dots, K$.
The decoder seeks to recover $X_0=f(X_1,\dots, X_K)$ with high probability. Do schemes exist with failure rate ...
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For positive $a$, $b$, $c$, $d$, if $\sum_{cyc}\frac1{1+a}=2$, (dis)prove $\frac{4(3\sqrt2-4)}{a+b+c+d}+\sum_{cyc}\frac1{\sqrt a}\geq3\sqrt2$
An open problem from Art of Problem Solving (AoPS):
If $a,b,c,d$ are positive real numbers such that
$$\frac{1}{1+a}+\frac 1{1+b}+\frac 1{1+c}+\frac 1{1+d}=2,$$then prove or disprove
$$\frac1{\sqrt a}...
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If there are $k_1, k_2, \dots, k_n$ divisions by $2$ in a Collatz cycle, then $k_1 + \dots + k_n \geq n$, but can we get a greater lower bound?
Let $f(x) = |3x + 1|_2(3x + 1)$ be the accelerated Collatz function, where $|3x + 1| = 2^{-\nu_2(3x + 1)}$ is the $2$-adic absolute value.
Clearly for all $x$ odd we have $\nu_2(3x + 1) \geq 1$ so ...
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The Collatz Conjecture function should induce a collection of Grothendieck groups, one for each $n \in \Bbb{Z}$ or $\Bbb{N}$. Their properties?
This question is about the Collatz conjecture.
Let $\Bbb{N}$ include $0$. The Collatz conjecture function is given by:
$$
f: \Bbb{N} \to \Bbb{N}, \\
f(n) = \begin{cases}
\dfrac{n}{2}, \text{ if } n = ...
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What is wrong with the following proof that $\{ (\frac{3}{2})^n\mod 1: n\in\mathbb{N} \} $ is dense in $\ [0,1]\ $?
What is wrong with the following straightforward proof that $\ \lbrace{ \left(\frac{3}{2}\right)^n \mod 1: n\in\mathbb{N} \rbrace}\ $ is dense in $\ [0,1]\ $ ?
In fact, let $\ f(x) = \lbrace{\ x^n \...
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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)=\...
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part V
(Preamble: This post is an offshoot of this MSE question and this MO question.)
My primary aim in this post is to compute a (hopefully factorable) expression for the quantity $n^2 - q^k$, if $N = q^k ...
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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)...
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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 $\...
2
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2
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what is so special about the number $3$ in the $3n+1$ conjecture?
Recently, I was quite intrigued by the $3n+1$ conjecture and it left me wondering what is so special about the number $3$? With the same rules apply, does there exist another positive integer other ...
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Open problems in Proof theory and Logic
There are numerous questions in the same form: "What are some open problems in mathematical logic". So for this we know:
Shelahs "Logical Dreams"
Logical Dreams
Friedmans "102 ...
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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 $\...
1
vote
1
answer
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What makes an unsolved problem "interesting"? [closed]
Beyond well-known unsolved problems like the Collatz conjecture or Recaman's sequence, one can trivially come up with problems in a similar vein (or perhaps an unusual infinite sum, or the ...
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Does this "improvement" to $I(m) < 2$, if $p^k m^2$ is an odd perfect number with special prime $p$, work?
(Preamble: This question is an offshoot of this earlier post.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, and the abundancy index of $x$ by $I(x)=\...
1
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1
answer
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II
The topic of odd perfect numbers likely needs no introduction.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
If $n$ is odd and $\sigma(n)=2n$, then we ...
3
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0
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On odd perfect numbers $p^k m^2$ with special prime $p$, satisfying $m^2 - p^k = 2^r t$
In what follows, we will denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. A positive integer $y$ ...