# Questions tagged [open-problem]

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

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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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### How relevant would it be to prove that P vs NP is equivalent to P vs NP using only machines with one letter input alphabet?

I was reading the official description of P vs NP at https://www.claymath.org/sites/default/files/pvsnp.pdf out of curiosity and the authot says "Does $\textbf{P = NP}$? It is easy to see that ...
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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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### 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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1 vote
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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 ...
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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 $y$ ...
1 vote
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### Three dimensional Hadwiger Nelson Problem

I am interested in Hadwiger Nelson Problem in higher dimensions. In particular, I have seen that the chromatic number for the Hadwiger Nelson Problem in three dimensions is between 6 and 15. But I ...
1 vote
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### Hadwiger-Nelson problem only on Q rationals

In the Hadwiger-Nelson problem, any two points unit distance apart must have distinct colors. However, it is known that if we restrict the vertices to only rational numbers, the chromatic number is ...
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### Longest-ever time between problem posing and solving?

Guiness World Record claims that Goldbach's conjecture is the oldest unsolved problem. A natural related question is what solved problem went unsolved for the longest time. In other words, of all the ...
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### Existence of continued fraction $\sqrt{n}$ with any period $k$

In this paper it is conjectured that for any positive integer $k$ there are infinitely many primes $p$ with the continued fraction expansion of $\sqrt{p}$ having length $k$ (Conjecture 5.1, https://...
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### Small question about the Lonely runner conjecture

I am currently looking into the lonely runner conjecture (Consider $k$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' ...
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### A combinatorial sieve for a specific setup

Let $F\subset\mathbb{Z}$ be a sifting set composed of $n\geqslant 2$ pairs of residue classes defined as $$F=\bigcup\nolimits ^{n}_{i=1}\{x:x\equiv \pm m_{i} \pmod{p_{i}}\}$$ where $p_{i}$ are ...
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### Is it a known result that the Collatz Tree repeats on itself indefinitely?

With reference to my visual pattern of the Collatz problem, I recognize that the Collatz tree for numbers of the form $6x-1$ (for $x\in\mathbb{Z}$ & $x>0$) and $1-6x$ (for $x\in\mathbb{Z}$ &...
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### Why is proving the Riemann Hypothesis so hard?

The Riemann Hypothesis is considered by many to be the most important unsolved problem in pure mathematics. Several attempts have been made in the last 150 years (here some of them are reported). RH ...
1 vote
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### Open questions in algebraic geometry that need insights from computations

I think that everything is in the question. I am looking for any open problems in algebraic geometry (vector bundles, Divisors, Frobenius morphism, counting points on Finite fields, Hitchin Fibration, ...
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### Has Any Currently Open Problem in Mathematics Definitively Been Shown to be Decidable?

There is a fairly extensive list of problems in various fields that have been shown to be undecidable. For example, see https://en.wikipedia.org/wiki/List_of_undecidable_problems And certainly, an ...
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### On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$ - Part II

(This question is an offshoot of this earlier post.) Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. Denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the sum of the ...
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### A Question Regarding Euler's Phi Function and Composite $N$

Let $N$ be a composite number. Do we know any necessary conditions on $N$ that will potentially allow $\phi(N) \mid N - 1$? Thank you.
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### Looking for (Previously Open) Problems That Have Been Resolved Contrary to Expectation

I am looking for reasonably famous mathematical problems that were once open for more than twenty five years (preferably more than a hundred) and whose subsequent resolution (proof or counterexample) ...
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1 vote
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### Number of unordered factorizations of a non-square-free positive integer

I recently discovered that the number of multiplicative partitions of some integer $n$ with $i$ prime factors is given by the Bell number $B_i$, provided that $n$ is a square-free integer. So, is ...
1 vote
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### What are the currently conjectures around symmetric group? [closed]

I am asking you this question: What are the currently conjectures around symmetric group on research? Indeed I am interested to work on unsolved problems concerning symmetric or alternating groups. ...
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### Is it an open problem? (Sums of squares) [duplicate]

Has it been proven that if n can be represented as a sum of squares of two rational numbers, then it can also be represented as a sum of squares of two integers? This is a relatively simple statement, ...
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### What is the current status of the snark theorem?

Wikipedia writes "W. T. Tutte conjectured that every snark has the Petersen graph as a minor... In 1999, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas announced a proof of this ...
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### Closed Nowhere dense is frentier of an open set

Excuse me can you see this question Every closed nowhere dense set is the frontier of an open set ... I tried on it but i am not sure , I prove it as follows Let $A$ is closed nowhere dense set, ...
So I'm interested in working on the Abundance conjecture, which states that for every projective variety $X$ with Kawamata log terminal singularities over a field $k$, if the canonical bundle $K_{X}$ ...