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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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Is this generalized form of Pillai's conjecture feasible?

In trying to design an algorithm for testing the equality of two power towers, I needed to assume the following generalized form of Pillai's conjecture: There is a constant $c$ such that there is ...
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Open conjecture in number theory without a good heuristic?

For most open conjectures in number theory , there are good heuristics that they are true : Examples include the Goldbach conjecture, the Collatz conjecture , the Riemann hypothesis and the twin prime ...
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"What is the specific mathematical reason behind the origin of the Collatz Conjecture that makes it difficult to solve it? [closed]

Is there a known so specific mathematical reason that makes it difficult to solve the Collatz Conjecture? Clearer: What is the specific mathematical reason behind the origin of the Collatz ...
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101 views

Why is the Collatz Conjecture so difficult to prove or disprove? [duplicate]

Is there some quality of the Collatz Conjecture that has made it so difficult to prove or disprove? Besides just using a computer to calculate lots and lots of values of $n$, of course.
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Constraint diophantine equations (open problem)

Given an odd number $N$ such that $N\equiv 3 \mod 4$, I am looking to find numerically or algebraically a positive natural number $k$ s.t, $1\leqslant k \leqslant \frac{N-3}{4}$, satisfying the ...
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Proof of Validity of My Polynomial Time Algorithm for $co-NP$ Complete Problem

I posted an algorithm yesterday, that purported to solve the co-NP Complete 'Boolean Tautology Problem' in polynomial time. Link to the algorithm : polynomial time algorithm In that post, I presented ...
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How is this not a proof of the Jacobian conjecture in the complex case?

I've just been reading the Wikipedia entry regarding the Jacobian conjecture, and it said that either the conjecture is true for all fields of characteristic zero, or it is false for all such fields. ...
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Which other unsolved problems, have necessary restrictions on the prime gaps?

We all know of Unsolved problems, like Goldbach,Legendre, and Grimm's conjectures. Goldbach has the necessary condition of: There exists a prime between $n$ and $2n-2$, which means prime gaps are ...
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What is currently the highest lower bound for the length of a nontrivial cycle in the Collatz Conjecture?

We know that there are two possibilities to disprove the Collatz Conjecture. We find a nontrivial cycle. We find a sequence that diverges to $\infty$ A non-constructive disproof is imaginable as ...
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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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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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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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$5 \times 5\;$ “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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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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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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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
107 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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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 ...
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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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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
115 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/...
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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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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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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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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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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
540 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 ...
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1answer
109 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
133 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 ...
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1answer
74 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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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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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 ...
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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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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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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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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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67 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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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 ...
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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 ...
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142 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)...
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1answer
135 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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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 ...
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1answer
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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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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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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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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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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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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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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_{...