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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2answers
103 views

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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1answer
104 views

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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44 views

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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1answer
77 views

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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3answers
143 views

$u_{n+1}=\frac{e^{u_n}}{n+1}$

One can prove that for $x\in \mathbb{R}$, the sequence $$ u_0=x\text{ and } \forall n\in \mathbb{N},\qquad u_{n+1}=\frac{e^{u_n}}{n+1} $$ converges to $0$ if $x \in ]-\infty,\delta[$ and diverges to $+...
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39 views

A Fundamental Theorem of Algebra type Result

Consider a polynomial $$p(t)=\sum_{i=1}^n a_it^{b_i}+a_0$$ of degree $b_n$, show that it admits at most $n+1$ nonnegative roots and at least one complex root. Assume all $b_i$ are positive and $a_i$ ...
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1answer
91 views

If $\Delta O \pmod {2^n} = (2)$ in $\Bbb{Z}/2^n$ for all $n \geq 1$, then does $\Delta O = (2)$ in $\Bbb{Z}$?

Let $O = $ the set of odd primes in $\Bbb{N}$. And $M = \Delta O = \{ x-y: x,y \in O\}$. Then we can take either set $X$ modulo $n$ for any $n \geq 2$: $\overline{X} = \{ x + (n) : x \in X\}$. Then ...
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135 views

About mapping class group.

What is best book for self learning mapping class group? I read "A Primer on Mapping Class Groups" By Benson Farb, Dan Margalit. Is there a topological space $X$ where we don't know $\...
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26 views

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 ...
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1answer
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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40 views

Set Cover Optimization of Cubes by Balls of Equal Radius

Let $C=[a,b]\subset R^n$ be a $n$-dimensional rectangle, $a,b\in R^n$. Find the minimal radius $r$ and set $x_1, \dotsc, x_N\in R^n$, for a given integer $N$ such that $$\bigcup_{i=1}^N B_r(x_i) \...
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2answers
88 views

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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47 views

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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Open Problems in Homological Algebra

This time I want to ask about open problems in : 1.- Homological Algebra. 2.- Cohomology of Groups. 3.- Algebraic Topology. 4.- Galois Cohomology. Is there a list ? My motivation : I must to present a ...
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1answer
137 views

What is the true status of the Lehmer totient problem?

The Lehmer-totient problem : For a prime number $\ n\ $ we have $\ \varphi(n)=n-1\ $. In particular, we have $\ \varphi(n) \mid n-1\ $. Is there a composite number $\ n\ $ with $\ \varphi(n)\mid n-1\ ...
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1answer
36 views

Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}$

Question: Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}?$ For instance, $\cos^2(z)+\sin^2(z)=1$ is still true when we move to complex ...
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4answers
169 views

What is the closest mathematical area related to the Collatz Conjecture?

I want to study mathematics at university. But, it is extremely important that the area I have chosen is correct. I need to know this to deal with the Collatz Conjecture, which is famous in ...
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67 views

Current Open Problems Similar to the Basel Problem? [closed]

Are there any current open problems that are similar to the Basel problem?
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40 views

Open problems related to Euler's phi function

Carmichael's conjecture is a relatively known problem related to Euler's totient function. I just found out about Lehmer's totient problem which is another problem related to this function. I would ...
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0answers
61 views

Searching for unsolved problems in the field of stability

I have proposed an approach for constructing Lyapunov functions for autonomous systems in my Ph.D. thesis and find some useful examples. Now, I am searching for some another example in this field or ...
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57 views

Reference Request : The number of binary string with some special condition (open problem)

(Sorry for my poor english...) I wonder the reference of Simon Marais Mathematics competition 2019 problem B4. This problem is as follow. (They said this problem is open problem.) B4. A set $\...
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58 views

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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1answer
44 views

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, ...
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1answer
80 views

Interpretation and Implications of the Abundance Conjecture

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 $K_{X}$ ...
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1answer
60 views

Open Problems to do with Polynomials and/or Elementary Function Theory

I was wondering what are some open problems (even if deemed impossible) in basic function theory (stuff you'd learn in high school) and/or open problems to do with polynomials... Thank you in ...
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1answer
87 views

Should I waste my time with wolfram alpha to solve such problem like $x^3+y^3+z^3=390$?

I'm sorry to ask this question probably it is not suitable here. However I read some papers in number theory related to the solution of $x^3+y^3+z^3=n$ for some known solved problem about ...
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1answer
48 views

Find all positive integers [closed]

Find all of non-null positive integers such that: $$(x-\frac{1}{yxz})\cdot (y-\frac{1}{yxz})\cdot (z-\frac{1}{xyz}) \in \mathbb{N}$$
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1answer
93 views

If $C_2$ is irrational, then there are infinitely many twin primes?

This is a natural follow-up after question 3629282. It is trivial that the irrationality of Brun's constant $B_2\approx1.90216$ implies that there are infinitely many twin primes: $$ B_2 \mbox{ is ...
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61 views

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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0answers
31 views

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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130 views

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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47 views

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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0answers
48 views

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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82 views

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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1answer
65 views

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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1answer
55 views

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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1answer
41 views

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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0answers
56 views

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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1answer
290 views

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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1answer
43 views

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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3answers
107 views

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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0answers
177 views

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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3answers
94 views

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 ...
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1answer
171 views

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 ...
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60 views

Proof verification upper bound on the Mondrian Art Problem

I have been doing some thinking on the Mondrian Art Problem and think I may have discovered something. I think I have improved the upper bound for odd numbers from $k$ (for a $k$ by $k$ square) to $(...
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63 views

Given $n$, exist $k$ such that $2^k$ contains $n$ as string. [duplicate]

I have this doubt: Given $n\in \mathbb{N}$, does exist $k\in \mathbb{N}$ such that $2^k$ contains $n$ as a string in it? For example, $53$ is in $2^{16}=65\color{red}{53}6$. I just thought the ...
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2answers
84 views

Are there any odd perfect numbers in base $g$, where $g \neq 10$?

The topic of odd perfect numbers likely needs no introduction. Here is my primary question: Are there any odd perfect numbers in base $g$, where $g \neq 10$? Of course, there are several ...

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