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

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1answer
39 views

Sum on digits of powers of two is not too large

Is the following proved: Are there infinitely many positive integers $m$ and an integer $n$ such that sum of digits of $2^m$ is at most $n$?
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0answers
78 views

Research Topics Needed

This coming academic year a professor has asked me to find some topics that I wish to pursue to write about. The problem/topic that will be discussed doesn't have to be open, but my trouble is that I ...
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24 views

Why the proof of Catalan's conjecture is not easily generalizable?

Let $x,y>0$, $u,v>1$ be integers. Why is it easier to solve $x^u-y^v=1$ than $x^u-y^v=2$? Is there possible some group behind the first equation which has some nice property that the group made ...
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1answer
76 views

Can one prove a special case of Goldbach conjecture without constructing primes?

I know that sometimes in mathematics one can prove that there exist something without constructing it. I was thinking whether one can show if $2^{57885162}$ is a sum of two primes by any reasoning. ...
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70 views

$q\sin q$ is small

I read from the book "Which Way did the Bicycle Go" that it is unknown whether for every $c>0$ there are infinitely many integers $n$ such that $|n\sin n|<c.$ Let $\mathbb{Q}_{m}$ be the set ...
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2answers
146 views

A Problem on Time Complexity of Algorithms

I want no know if the following problem is solved or not, or how can I solve it? Problem: For every integer $t$, Is there any problem that can be verified in $O(n^{s})$ but its solution can be found ...
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0answers
64 views

Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is ...
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0answers
49 views

Strict positivity on the diagonal of a particular integral kernel: A question from Simon's Schrödinger Semigroups

This is a question pertaining to a (formerly?) open question from Barry Simon's Schrödinger Semigroups. In Theorem C.5.2 (page 504) of that publication, the existence of a specific function ...
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0answers
191 views

Open problems with practical outcome in a visible future?

[Note] The question has been asked at Matheoverflow. But there is no answers. I believe that any non-trivial idea will sooner or later find application in real life. However "sooner" is better than ...
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0answers
134 views

Panel structure on epi F**

It is known that, generically, the convex hull of a hypersurface embedded in $\mathbb{R}^n$ has a panel structure of simplices. Of course one can construct embeddings where this is not the case, but ...
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0answers
213 views

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable? separable = $X$ has a countable dense subset. A space $X$ has a zeroset-diagonal when there is a ...
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2answers
2k views

What is the most important unsolved mathematical problem left? [closed]

Some time ago Andrew Wiles proved fermat's last theorem. The four colour theorem has been proved and Kepler's Conjecture has been proved. But what is the most important mathematical proof yet to be ...
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1answer
185 views

Proving that $\sum_{k=n}^{2n-2} \frac{|\sin k\ |}{k} < 0.7\ln 2$ for $n\ge2$

Prove that $$\sum_{k=n}^{2n-2} \frac{|\sin k|}{k} < 0.7 \ln 2, \qquad (n\ge2)$$ and $$\cot\left(\frac{\pi}{2n}\right) \le \sum_{k=1}^n \left|\sin\left(x+\frac{k\pi}{n}\right)\right| \le ...
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1answer
239 views

Waring's Inequality Solution

$$ \text{Waring's problem asks, "Is }\left\lfloor \left(\frac{3}{2}\right)^n\right\rfloor =\left\lfloor \frac{3^n-1}{2^n-1}\right\rfloor\text{ always true?"} $$ We craft an inequality, with $m,n ...
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1answer
130 views

Catch stochasticity of nature

Do you know of anything that comes close to topic 3 on http://www.darpa.mil/Our_Work/DSO/Programs/23_Mathematical_Challenges.aspx Capture and Harness Stochasticity in Nature Address Mumford’s ...
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1answer
305 views

symmetric difference of languages - both are in NP and coNP

I have this problem, Let $L_1,L_2$ be languages in $NP \cap co-NP$. I want to show that their symmetric difference is also in $NP \cap co-NP$. Like: $L_1 \oplus L_2 = \{x | x$ is in exactly one of ...
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1answer
74 views

How can I solve this problem without doing it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
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1answer
13 views

Dickson's (and Bunyakovsky's) conjecture with compositeness constraints

Dickson's conjecture, in simple terms, says that for any choice of $a_1,b_1,a_2,b_2,...,a_k,b_k\in\Bbb N$ we have, for infinitely many $n\in\Bbb N$, that all of $a_1+nb_1,...,a_k+nb_k$ are prime, ...
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0answers
14 views

Does Schinzel's hypthesis hold when you allow exponentials?

I'm curious to what extent Schinzel's hypothesis is expected to hold. Heuristically it seems that a typical doubly exponential function say $3^{3^n}+4$ probably is prime only at finitely many $n$ by ...
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1answer
25 views

NP-Completeness and NP

Given : $S$ is an $NP-Complete$ problem $Q$ and $R$ are two other problems not known to be in $NP$. $Q$ is polynomial-time reducible to $S$ and $S$ is polynomial-time reducible to $R$. My thoughts ...
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18 views

Dominance Network Worded Problems

What are some methods to solve this? Normally for dominance I do as such: Write a matrix for one step dominance, then find total dominance by = D+D^2 - then sum each row of the matrix. Using this ...
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1answer
42 views

Problem Solving - Project Crashing Time

My working out: (EST,EFT) times for the activities: A: (0,0) B: (0,8) C: (3,3) D: (10,38) E: (10,18) F: (18,18) G: (25,33) H: (58,58) I: (25,33) J: (45,53) K: (118,118) Finish: (133,133) ...
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18 views

an idea about the relation between the residues at infinity and the infinity line?

i have a question about the following three concepts for infinity: The point at infinity, also called ideal point, of the real number line is a point which, when added to the number line yields a ...
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73 views

Conjecture related to the Erdős discrepancy problem

Conjecture: If $k \in \mathbb{N}$ and $S$ is an infinite set of primes, then the multiplicative $\pm$-sequence generated by $S$ contains $+^k$ as a substring infinitely often. (If $S$ is allowed to ...
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1answer
89 views

Perfect cuboid cube

Is there any proof that there is no cubic perfect cuboid? Here is a description of the problem: . I'm currently using trying to get an empty set to solve it... [ A "perfect cuboid" is one whose ...
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1answer
60 views

will $x_{n+1}=x_n/2$ if $x_n$ is even; otherwise $x_{n+1}=3*x_n+1$, will $x_n$ shrink to 1?

I was asked this question that, for any $x_1 \in \mathbb{N}$, define the sequence as $$x_{n+1}=\left\{ \begin{array}{l l} x_n/2 & \quad \text{if } x \text{ is even} \\ 3 x_n+1 & ...
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0answers
141 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
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53 views

Reference for the cardinality equal to continuum of space

Recently, I am considering an open problem, see here. Now, I want to prove the answer to the problem is true when the case that the cardinality of topological space is equal to continuum. However, I ...
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1answer
92 views

a Problem about Sequence [duplicate]

Let $a_1$ be an integer. Then we assume $$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$ Now we prove that for any ...
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2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...