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)\...
Danka Makabre's user avatar
1 vote
0 answers
40 views

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,\...
Marco Ripà's user avatar
2 votes
2 answers
116 views

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 ...
user107952's user avatar
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2 votes
1 answer
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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 ...
Adam Rubinson's user avatar
2 votes
0 answers
166 views

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 ...
Bertrand Haskell's user avatar
0 votes
1 answer
57 views

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 ...
it's a hire car baby's user avatar
2 votes
2 answers
132 views

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 ...
Bertrand Haskell's user avatar
0 votes
0 answers
66 views

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 ...
Enhao Lan's user avatar
  • 5,783
59 votes
27 answers
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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 ...
2 votes
1 answer
88 views

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 ...
MathCrackExchange's user avatar
0 votes
0 answers
32 views

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 ...
slithy_tove's user avatar
0 votes
0 answers
30 views

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 ...
Daniel's user avatar
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1 vote
0 answers
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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),$...
William Zheng's user avatar
8 votes
0 answers
162 views

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 ...
uhoh's user avatar
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8 votes
1 answer
117 views

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 ...
Zoe Allen's user avatar
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3 votes
0 answers
139 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, $...
MathCrackExchange's user avatar
-2 votes
1 answer
50 views

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 $H'$...
MathCrackExchange's user avatar
0 votes
2 answers
113 views

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 ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
1 answer
69 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 ...
Marco Ripà's user avatar
1 vote
0 answers
83 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 ...
Marco Ripà's user avatar
3 votes
1 answer
249 views

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 ...
L. E.'s user avatar
  • 399
4 votes
1 answer
99 views

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 ...
TheBestMagician's user avatar
3 votes
0 answers
68 views

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 $...
A R's user avatar
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3 votes
0 answers
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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 ...
maac's user avatar
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1 vote
0 answers
63 views

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 ...
Akira's user avatar
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0 votes
1 answer
63 views

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 $\...
Snacc's user avatar
  • 1,860
0 votes
1 answer
66 views

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 ...
MathCrackExchange's user avatar
0 votes
1 answer
65 views

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-...
Synthels's user avatar
  • 151
0 votes
2 answers
113 views

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 ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
124 views

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 ...
user759001's user avatar
2 votes
1 answer
63 views

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 $\dim(Y)<\infty$ $\...
F.R.'s user avatar
  • 168
6 votes
1 answer
425 views

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 ...
Tanaka's user avatar
  • 125
3 votes
1 answer
148 views

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\} $ ...
Adam Rubinson's user avatar
3 votes
3 answers
780 views

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 ...
profPlum's user avatar
  • 327
1 vote
1 answer
185 views

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 ...
Christian Chapman's user avatar
11 votes
0 answers
566 views

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}...
Will's user avatar
  • 135
0 votes
2 answers
188 views

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 ...
MathCrackExchange's user avatar
7 votes
1 answer
230 views

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 = ...
MathCrackExchange's user avatar
2 votes
1 answer
196 views

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 \...
Adam Rubinson's user avatar
1 vote
0 answers
136 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)=\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
0 answers
58 views

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 ...
Jose Arnaldo Bebita Dris's user avatar
0 votes
1 answer
137 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)...
Jose Arnaldo Bebita Dris's user avatar
0 votes
2 answers
274 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 $\...
Jose Arnaldo Bebita Dris's user avatar
2 votes
2 answers
206 views

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 ...
UnsinkableSam's user avatar
3 votes
0 answers
319 views

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 ...
LogicTheorist's user avatar
0 votes
1 answer
111 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 $\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
228 views

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 ...
BADSAP's user avatar
  • 177
1 vote
0 answers
36 views

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)=\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
422 views

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 ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
0 answers
222 views

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 $y$ ...
Jose Arnaldo Bebita Dris's user avatar

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