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Questions tagged [open-problem]

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

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Heesch numbers in 3D

At the Tiling Database there are 3, 20, 198, 1390 (A054361) non-tiling polyominoes of order 7 to 10. In 3D, solids with a particular Heesch number don't seem to be well known. Glenn Rhoads found ...
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0answers
19 views

Balanced weight perfect matching

Given an undirected graph $G = (V,E)$, edge weight $w_e \ \forall e \in E$, I'm interested in the following problem. Find a perfect matching $M \subseteq E$ that minimizes $(\max_{e \in M} w_e - \...
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45 views

Open problems on geometrical shapes

I'm looking for simply-stated open problems requiring the optimization of 2D geometrical shapes. The moving sofa problem would be an example. Does anybody know of a good source for these?
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0answers
330 views

$5 \times 5\;$ “square additive set”

Problem: IBM Research - Ponder This - January 2019 monthly contest (which was closed few days ago) leads to the problem: Find sets $A = \{a_1,a_2,\ldots,a_n\}$, $B = \{b_1,b_2,\ldots, b_m\}$ such ...
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4answers
137 views

What mathematical consequences might there be if Euler Mascheroni constant is rational?

So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant. Since we cannot prove ...
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0answers
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Does the Graceful tree conjecture refer to all or only some trees?

I am a total beginner in this field and i‘m not really versed in the terminology, so please bare with me. What I know, is that a graceful labeling, refers to a tree with $n$ vertices, where each ...
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70 views

I need a help with my research. Just asking for inventive answer to my question. [closed]

Let's take a value and give it to x. Let's say x has its growth doesn't matter how much. Let's describe his growth on a number scale like a man on a boat going in a direction of a river flow. Let's ...
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0answers
24 views

boolean logic minimization

Two level boolean logic minimization(AND, OR, NOT) for big number of variables,(say n) is time-consuming. I want to minimize an n variable sum of Product form boolean expression efficiently where n ...
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1answer
94 views

Where is the flaw in this proof of Legendre's Conjecture?

Introduction The following argument has been advanced by one of my friends which attempts to prove the Legendre's Conjecture. I could find no flaw in the argument and so I am posting it here in the ...
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16 views

counting simple path transitions between an incomplete graph and it's complement

Now, I know that counting cycles of an incomplete graph is generally considered to require exponential time, however I am wondering if there is a formula for this special case (I am actually going to ...
3
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2answers
93 views

Elements of infinite order in CAT(0) groups

In E. Swenson, A cut point theorem for CAT(0) groups, J. Differential Geom. 53 (1999), no. 2, 327–358. the author shows (Theorem 11) that if a group $G$ acts geometrically (i.e. properly ...
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53 views

Probability of a coin of $1$ cm diameter landing completely on a square cloth of side $2$ cm?

I have a coin of diameter $1$ cm, and I toss the coin over a square cloth of 2 cm by 2 cm. Find the probability that the coin lands completely on the cloth i.e. no part of the coin lies outside the ...
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1answer
87 views

Intersection of polytope and hyperplane

Determining the intersection of polytope and hyperplane in arbitrary dimension is of central interest in computational geometry. In some paper(this one in particular: https://pdfs.semanticscholar.org/...
5
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1answer
120 views

Is there any function whose limit at $x_0$ is unknown?

I would like to know if there is any non trivial function $f(x)$ and a $x_0$ such that $$\lim_{x\to\ x_0} f(x)$$ is currently not known, with $x_0 \in \mathbb{R}\cup \{-\infty, +\infty \}$. An example ...
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64 views

Conjectures for which there are strong heuristic arguments both for and against

I find the conjecture that there are infinitely many Fermat primes to be very interesting, because there is a very reasonable-seeming and semi-quantitative heuristic argument for the conjecture, but ...
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1answer
42 views

Is there a semi-decidable statement equivalent to the Collatz-conjecture?

We cannot rule out that the Collatz-conjecture cannot be proven. But we also cannot rule out that it is false and we cannot prove this in the case the sequence diverges for some start-number. Is ...
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1answer
190 views

Does the list of “number of groups of order $n$” contain every natural number?

In other words: For every natural number $m$, does there always exist an $n$ for which there are exactly $m$ groups of order $n$ up to isomorphism? Or is this an open question in mathematics? If ...
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2answers
82 views

Currently open questions that would not require infinite memory to evaluate computationally

I was reading an old thread over at the code golf / code challenge Stack Exchange, which asks users to come up with a computer program for which it is an open question whether it terminates. My first ...
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1answer
388 views

Why $9k \pm 4$ can't be written as sum of four cubes

All integers except possibly those of the form $9k \pm 4$ can be written as the sum of 4 cubes. Are there any heuristics that suggest this is impossible in general, or is it the case that no one ...
4
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1answer
106 views

Disprove $m!=100x^2+20x$ using an estimation for factorial.

$\newcommand{\floor}[1]{\lfloor #1 \rfloor}$ I have the equation $m!=100x^2+20x$ where $x$ and $m$ are real non-negative integers. I wish to disprove for when $m\geq20$ how can I do this? I had an ...
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1answer
126 views

Is $n^2 - n + 1$ prime infinitely often?

It seems that this is an open question, though I haven't been able to find many references pointing to work towards the answer. I know that $n^2-n+1$ is conjectured to be prime infinitely often by ...
2
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1answer
72 views

Can we rigorously define certainties of famous conjectures?

Is there some precise probabilistic framework to define levels of certainty of mathematical conjectures given current mathematical knowledge? If so, what would be the certainties attached to ...
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0answers
87 views

Are there infinite many positive integers $n$ such that $n^2 + n +1$ is prime?

I've heard that linear polynomials with proper integer coefficients has infinite many positive integers $n$ such that $f(n)$ is prime, by Dirichlet's theorem. But is there something done with ...
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0answers
101 views

Has it yet been proven that $x_{n+1}=x_n-\frac{1}{x_{n}}$ with $x_0=2$ is unbounded?

Concider the following recursive formula: $$x_{n+1}=x_n-\frac{1}{x_{n}}$$ with $x_0=2$. Is the sequence produced by this formula unbounded? Some time ago I came across this problem somewhere on the ...
3
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1answer
94 views

Prove or Disprove: $e = m\pi + n$ for some integers $m$ and $n$. [closed]

I would like to prove or disprove the following statement: There exist integers $m$ and $n$ such that $e = m\pi + n$ (where $e=2.7...$) Edit: any pointers would be appreciated.
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1answer
87 views

Are there any false variants of the Collatz conjecture for which the probability heuristic works?

One of the supporting arguments for the Collatz conjecture is the probability heuristic, which states roughly that because the collatz operations tends to decrease numbers over time, it probably doesn'...
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2answers
65 views

How to write a proof? Use English or mathematical notations? [closed]

I am now considering writing a mathematics paper. One thing that bothers me a lot is the following question How far should I use notations instead of English sentence? On one hand, it is easy for ...
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1answer
28 views

A random geometric-type series (which is not strange)?

Let $\psi_n$ be a sequence of uniformly distributed IID random variables taking values in the $interval$ $[0,1]$. What is the probability that $\sum_0^\infty (\psi_n)^n$ is converging? (Note: $()^n$ ...
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1answer
65 views

Density of product of primes plus 1? [closed]

Let $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11$, and $p_n$ be the n-th prime number. The numbers $a_n:=p_1p_2p_3...p_n+1$ are well known from showing that $a_n$ is not a finite set. However, I am intersted in ...
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78 views

Why is this number : $e^{e^{e^{79}}}$ conjectured to be an integer number which is a skew number? [duplicate]

skew number is defined as : $e^{e^{e^{79}}}$ , I seek for the mathematical reasons which let $e^{e^{e^{79}}}$ conjectured to be an integer ?
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Conjectured values for infinite products?

When I look at simple open problems in calculus ( not including analytic Number theory mainly ) most conjectures are , or are equivalent to , infinite sums or integrals being equal to a closed form ...
4
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1answer
212 views

Do $e$ and $\pi$ satisfy a non-trivial diophantine equation?

Do there exist integers $a,b,c$ such that at least one of them is non-zero and $ae+b\pi = c$? Please note that, I am only asking if e and $\pi$ can satisfy a non-trivial diophantine equation $ax + by ...
2
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1answer
140 views

Divergent $3n+1$ sequence?

Recall the Collatz function given by: $$ T(n) = \begin{cases} {\dfrac{n}{2}} & n \equiv 0\pmod 2\\ & \\ 3n+1 & n \equiv 1\pmod 2 \end{cases} $$ The well-known conjecture states that $T^{(k)...
2
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1answer
124 views

Lonely Runner Conjecture proof for $k=3$ runners

I'm currently reading Lonely runner conjecture which has been proved upto $k=7$. I could understand the cases with $k=1,2$ i.e, one and two runners repsectively, but I am stuck at $k=3$. Can anyone ...
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2answers
262 views

Is the Erdős–Faber–Lovász conjecture open still?

Is the Erdős–Faber–Lovász Conjecture open still? According to Wikipedia it is unsolved still, but I think this is not hard to solve this conjecture. Conjecture: If $n$ complete graphs, each ...
2
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1answer
79 views

Are mersenne numbers with prime exponent cube free?

It is not known if mersenne numbers with prime exponent are square free. It is an open problem in number theory. Some limitations on the divisors are discussed in: https://mathoverflow.net/questions/...
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100 views

Reference to the Hodge conjecture

My knowledge on Hodge conjecture is very low. I want to start to study related topics to the Hodge conjecture . Is it possible to understand this conjecture by differential geometry and without ...
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0answers
87 views

Theorems in the distribution of the primes without elementary proofs

From the turn of the 20th century, the thought of an elementary proof of the prime number theorem was the obvious holy grail for elementary methods, but after the work of Selberg and Erdos in the 40s ...
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119 views

2-in-3-SAT vs. Gaussian elimination

$\mathsf{P}\stackrel{?}{=}\mathsf{NP}$ is an open question currently. However monotone 2-in-3-SAT is an $\mathsf{NP}$-complete problem. Further any 2-in-3 clause $(x\lor y\lor z)$ can be reduced to a ...
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0answers
127 views

Euler Totient of Numbers Between Twin Primes.

Are there any known special properties of a number located between twin primes? The question came up in the discussion. (The expression below has been rephrased to a weaker form for clarity) In ...
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0answers
105 views

k-tuple Admissibility?

Admissibility is defined for the integer $k$-tuples by the non-existence of a prime number $p$, for which each element of the $k$-tuple under union forms a complete residue system $\pmod{p}$. If we ...
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2answers
396 views

Compute $S_n=\sum\limits_{a_1,a_2,\cdots,a_n=1}^\infty \frac{a_1a_2\cdots a_n}{(a_1+a_2+\cdots+a_n)!}$

It is tagged as an open problem in the book Fractional parts,series and integrals. If this proof is valid , I don't have any idea how to get it published so I posted it here . $\displaystyle \sum_{...
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4answers
1k views

Polynomials $f$ and $f'$ with all roots distinct integers

Edit 2. Since the question below appears to be open for degree seven and above, I have re-tagged appropriately, and also suggested this on MathOverflow (link) as a potential polymath project. Edit 1. ...
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0answers
198 views

Is the Traveling Salesman Problem with Precedence Constraints NP-hard?

I am searching for a proof of NP-hardness or dynamic programming solution to the Traveling Salesman Problem with Precedence Constraints (TSP-PC). So far I could not even find any proof that proves the ...
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0answers
146 views

General mathematical consensus on the correct answer to each Millenium Prize Problem

This question is an extension of Open mathematical questions for which we really, really have no idea what the answer is, although it may immediately get closed as vague and primarily opinion-based. ...
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1answer
116 views

Can we prove that there are no two perfect powers with difference $6$?

Here : Are those lists known to be complete? I asked whether the list of numbers given in the link is known to be complete. Since the generalized Catalan-conjecture is apparantly open, I think they ...
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0answers
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Who knows a survey about the status of the generalized taxicab-numbers?

Here : https://en.wikipedia.org/wiki/Generalized_taxicab_number the generalized taxi-cab-numbers are mentioned and it is stated that no positive integer is known to be expressible by a sum of two ...
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1answer
29 views

Pebbling number of the grid $P_m\square P_n$

I looked around for some while but couldn't find anything. Is the pebbling number known for the grid $P_m\square P_n$? (I'm not asking about the optimal pebbling number.)
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2answers
137 views

Find all numbers whose factorial equals the product of (more than 1) consecutive numbers greater than that number.

This problem is basically equivalent to the nontrivial case of the problem of finding a product of two factorials that is also a factorial, but that problem seems to be open as well. The keyword here ...
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0answers
166 views

Property of Solutions to Two Correlated Optimization Problems

Here are two integral equations that correlated with each other For $w\in[0,w_0]$: \begin{equation} x^*_w\triangleq\begin{cases} &-\infty,~\mbox{if }\nexists~x\geq 0~\mbox{such that }y^*_v+p-v\...