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

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Open problems in Banach spaces? universal spaces

I have gathered a list of universality problems in Banach spaces which have been solved: The non existence of a separable reflexive space universal for the class of separable reflexive spaces. If a ...
2
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0answers
85 views

Big list of references [divided by categories] that collect commented open problems and conjectures [closed]

The aim of this question is to collect a big list of books or survey papers or websites which collect an up-to-date, comprehensive, well-organized, and possibly commented list of open problems. I ...
2
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1answer
82 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This has been cross-posted to MO.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). ...
2
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0answers
57 views

What do we know about Sorli's Conjecture and Odd Perfect Numbers?

I have seen that Jose Arnaldo Dris has recently published a great deal of literature concerning odd perfect numbers and Sorli's Conjecture. He claims that the truth of Sorli's conjecture implies that ...
2
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0answers
106 views

A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
2
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106 views

When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333, 333, 333, 333, 333, 333, 333, 333, 334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ are ...
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0answers
50 views

Apery's constant

I read that it is unknown if $\zeta (3)$ is algebraic but it is known to be irrational. Has anyone proved anything of the form $\zeta (3)$ is not a root of a polynomial of degree $12345$ with integer ...
2
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0answers
109 views

Are there infinitely many primes $p_k$ such that $(p_k-1)!+p_k$ is a also prime?

I am wondering whether there are infinitely many primes $p_k$ such that $(p_k-1)!+p_k$ is also prime. Given that $p_k \equiv 2 \pmod 3$. For a very large prime, I can assume Stirling's ...
2
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263 views

Invariant Subspace Problem

Louis de Branges had a paper on his homepage claiming a solution for the Invariant Subspace Problem. But I don't see that paper anymore, though he still has a "proof" of Riemann Hypothesis on his ...
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1answer
187 views

What is this variation of the traveling salesman problem called?

I've encountered a description of a well-known, hard to compute problem. Let us consider $n$ sites that need to be connected in the shortest possible way. If I am only allowed to connect the sites I ...
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2answers
98 views

How can I solve this problem without having to do it by hand?

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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2answers
88 views

Is there any way to solve this problem without having to do 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. Is there any way to group ...
1
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1answer
166 views

Collatz conjecture and related problems - mathematical machinery

Collatz conjecture stands as an open problem. That leads me to believe that the conjecture cannot be resolved by elementary means. Which brings me to my question: What techniques/machinery from ...
1
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1answer
256 views

Cube nets hexomino tilings.

I am looking for an ~12x12 rectangle (small holes and small obtrusions are okay) made entirely of cube net hexominos. It is my understanding that perfect rectangles, in general, are not possible ...
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1answer
57 views

Does proving the following statement equate to proving the twin prime conjecture?

After some research, I found that it has been supposedly proven, that proving that there exists an infinite number of positive integers K such that; $K \neq 6ab \pm a \pm b$ and $K \neq 6ab \mp a ...
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3answers
433 views

Tiling with polyominos

How to prove or disprove that if a polyomino tiles the plane, it must also be able to perfectly tile some larger polyomino, which also tiles the plane? A polyomino is finite set of unit squares ...
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0answers
49 views

Open problems for which all cases except one have been solved

Keller's conjecture states that in any tiling of Euclidean $n$-space by identical hypercubes there are two cubes that meet face to face. The conjecture has been shown to be true for $n<7$ and ...
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1answer
46 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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108 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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0answers
26 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 ...
1
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1answer
87 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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73 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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0answers
167 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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2answers
149 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
76 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
50 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
206 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
138 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
214 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
61 views

Is there a link between the Bunyakovsky conjecture and the Twin Prime conjecture?

Can the proof of one conjecture be considered a proof of the other conjecture? The general method of building an infinite number of prime producing quadratic polynomials was given in the link ...
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1answer
188 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
246 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
137 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
326 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 ...
0
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1answer
75 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 ...
0
votes
0answers
23 views

A detail in [ the time evolution operator]

If $$\hat C=\Delta \hat S_{22}+\lambda _1[\hat a_1 \hat S_{21}+\hat a_1^\dagger \hat S_{12}]+\lambda _2[\hat a_2 \hat S_{32}+\hat a_2^\dagger \hat S_{23}] ,$$ $$ \hat \beta= ...
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33 views

diffusion- stuck

In a round room of radius R, a large number of coins N of diameter d are randomly dispersed upon the floor. A ladybird starts from the centre of the room, crawling at speed v. Suppose that every time ...
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35 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
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1answer
18 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
21 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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0answers
32 views

Is there a finite set comprising the solutions to indefinite integrals of common functions?

There are some integrals that are impossible to express in terms of elementary function, for example, $ \int \frac{e^x}{x} dx $ is only expressible as a "special" function $Ei(x)$, the exponential ...
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1answer
26 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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1answer
104 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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87 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
116 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
63 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
54 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
93 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
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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 ...