# Questions tagged [open-problem]

Questions on problems that have yet to be completely solved by current mathematical methods.

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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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### 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 ...
1 vote
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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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### 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 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 ...
### 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 $y$ ...