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

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148 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 ...
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1answer
227 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
50 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
407 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
44 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
73 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 $\gcd(q, n) = 1$). ...
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0answers
28 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
41 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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86 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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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 ...
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1answer
83 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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71 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
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
67 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
198 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
135 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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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
186 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
241 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
313 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
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 ...
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32 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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16 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
15 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
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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20 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
54 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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75 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
96 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
61 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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147 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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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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Open problems of Clay Mathematics Institute. [on hold]

Mathematics research at the end of the $20$th century assumed unfathomable proportion as compared to the situation at the end of the $19$th century. The Clay role is to further the legacy of Hilbert's ...
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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 ...