# Questions tagged [open-problem]

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

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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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### Unsolved problems about roots of polynomials

I'm trying to find a list of conjectures about roots of polynomials that are easy to state but hard to prove. For examples, I know two such conjectures which are quite famous: (Sendov's conjecture) ...
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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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### Covering of Unit Square by Circles of Equal Radius

Let $N$ be fixed, find the minimal radius $r>0$ and the according set $x_1,\dotsc, x_N \in \mathbb{R}^2$ such that $$[0,1]^2 \subset \bigcup_{i=1}^N B(x_i,r),$$ with $B(x_i, r)$ being the closed ...
121 views

### 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 ...
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### Status of Deligne's Weight-Monodromy Conjecture?

I'm interested in Deligne's weight-monodromy conjecture, and was wondering if anyone could provide any insight on its current status... What is completely resolved? What is being worked on and is ...
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### Open questions in Probability

I was wondering if there are any really important or widely studied open problems in probability, we all know about the millennium problems list, but no one of those is directly related to probability....
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### Proof that the twin prime constant is irrational (reference request)

It seems intuitively clear that the twin prime constant $$C_2 = \prod_{p > 2} \left(1-\frac{1}{(p-1)^2} \right) \approx 0.66016$$ is irrational. Can anyone please give a reference to a short ...
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### list of open conjectures which are solved by students in high-Middle school level or receprocally?

I know only Snevily's conjecture which it was proved in 2009 by Bodan Arsovski. and He was a high-school student at the time., Are there other conjectures which are solved by students in high ...
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### If $q^k n^2$ is an odd perfect number with special prime $q$, then $k \neq 1$ (MSE version)

Note that $\gcd(\sigma(q^k),\sigma(n^2))=i(q)=\gcd(n^2,\sigma(n^2))$ if and only if $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$. I have therefore undeleted this question, in view of this closely ...
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### Could be in mathematics an open problem solved by another open problem ?if yes any example?

In mathematics there are many open problems or open conjectures some of them had been solved and others are still open , it is good to know if there exist an example of open problem wich is solved by ...
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### Open problems in Functions of One Complex Variable?

I am interested in complex variables, and only have knowledge for one variable, not several as of now... Regardless, I want to participate in some research... Do you guys know of any interesting open ...
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### Chromatic number of plane and sphere [closed]

I've been studying Hadwiger–Nelson problem and several questions arose: 1) Is it easier to get an answer (chromatic number) for plane with different definition of distance? I mean, if we consider ...
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### Possible relationship between non-divisors of odd perfect numbers and coefficients of corresponding cyclotomic polynomials?

A positive integer $n$ is called perfect if $\sigma(n)=2n$, where $$\sigma(n)=\sum_{d \mid n}{d}$$ is the sum of divisors of $n$. If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect ...
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### What does Chowla's cosine problem actually ask to find?

The problem according to this description is: Let $A \subseteq {\mathbb N}$ be a set of $n$ positive integers and set $m(A) = - \min_x \sum_{a \in A} \cos(ax).$ What is $m(n) = \min_A m(A)$? ...
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### Mathematical Consequences of $P=NP$ or $P\neq NP$ [closed]

It is common in mathematics to assume some unknown hypothesis or conjecture to be true or false, and then prove results dependent on said assumption. This can lead to many interesting developments ...
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### Updated: Can someone have a look at a simple attempt towards a direct mathematical proof of the Collatz conjecture?

I am looking for feedback on the below, any is appreciated :) Remark: Apologies in advance for any incorrect use of notation as my mathematical experience is quite novice, additionally a word of ...
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### Series sum of Fractional Fibonacci Series

Please help me proving that $\sum_{n=1}^{\infty} F(n)/(10^{-(n+1)}) = 0.011235955...$ Where F(n) are the Fibonacci Numbers
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### Are there any sets where it is an open question whether the set is open (or closed)?

Proving whether a set is open or closed in a topological space can be difficult at times, so I'm curious if there are any sets where it is an open question whether they are open or closed. Obviously, ...
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### Explicit results for the Collatz Conjecture? [closed]

I have some short questions I'd like to ask. I hope I'm asking in the right place. Has there been any significant progress on the Collatz Conjecture to date? What Tao really did was to set new ...
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### Question with eight eight numbers [closed]

Suppose we have eight $8$ and we have to reach $1000$ with eight $8$. Use $+ \space - \space \times \space \div \space \sqrt{} \space \cdots$ to reach $1000$. Please help me to find at least one ...
### What is unknown about the sphere $S^2$?
The standard sphere $S^2$ is (arguably) the simplest symmetrical geometric object. We can view $S^2$ as a smooth manifold in the category of smooth manifolds, or a Riemann sphere $\mathbb{P}^1$ in the ...