Questions tagged [open-problem]
Questions on problems that have yet to be completely solved by current mathematical methods.
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T/F: If $\alpha\in\mathbb{R}_{>1}$ is not Pisot, then $\{\lfloor\alpha^n\rfloor:n\in\mathbb{N}\}$ contains infinitely many odd and even integers
True or false:
If $\alpha\in\mathbb{R}_{>1}\setminus\mathbb{N},$ then
the set $\{ \lfloor \alpha^n \rfloor: n\in\mathbb{N} \}$ contains
infinitely many even integers and infinitely many odd ...
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Collatz sequence but multiplying by a large odd number rather than $3.$ What is the simplest way to prove that a sequence goes off to infinity?
Under the Collatz rules:
$n\to 837n+1$ if $n$ is odd
$n\to n/2$ if $n$ is even.
What is the simplest argument/proof to show that there is a Collatz sequence with a starting number that goes off to ...
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The $27$ dots problem
Here is a generalization of the well-known Nine dots puzzle to $3$ dimensions, where we introduce a new constraint (it is just a particular case of a more general problem that I have recently shared ...
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What's The Minimum Number Of Prime Factors Needed To Replace "3x+1" With Any Linear ("mx+b") Function And Still Work Like The Collatz Conjecture?
Apologies; I know there are a few assumptions used to pose this question, namely:
1): That yes, any mx+b function can work like the infamous "3x+1," problem...
...Provided, that you give it ...
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Implications of having access to the Busy Beaver oracle
Apologies if I'm asking a naïve question as I've only recently learned about the concept.
What would be the practical implications (if any) of having access to a magical black box providing the ...
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The n-th number open problems
Some open problems in mathematics boil down to the question of defining the $n$-th term of a certain sequence for a specific $n$. For instance, the value of the $5$-th diagonal Ramsey number and the $...
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For each integer $k,$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},$ s.t. for each $n,\ p_{n+1}=2p_n- 1$ or $p_{n+1} =2p_n+1?$
For each $k\in\mathbb{N},$ does there exist a $k-$tuple of primes,
$(p_n)_{n=1}^{k},\ $ such that for each $n,$ the following is
satisfied: $p_{n+1} = 2p_n- 1\ $ or $p_{n+1} = 2p_n + 1?$
If yes then ...
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The convergence of the Flint Hills series vs the convergence of $\lim_{n\to\infty}\frac{1}{n^3\sin^2(n)}$
The Flint Hills series, is the series $$\sum_{n=1}^\infty\frac{1}{n^3\sin^2(n)},$$ and it's an open problem as to whether the series converges. From the proof of Corollary 4 of this paper, it seems ...
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Are there any prime numbers $\geq 5$ which are not a factor of some $n!-1,\ $ where $n\geq 2$?
For each $n\in\mathbb{N},$ let $S_n$ be the set of prime factors of $n! + 1$. By Wilson's theorem, we have $\ p\mid (p-1)!+1\ $ for every prime $p.$ Therefore, $\displaystyle\bigcup_{n=1}^{\infty} S_n ...
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Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $
Background
The Norwegian mathematician and astronomer Carl Størmer did important work on the equation
$$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$
...
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Can we really be sure that there is no odd perfect number below $10^{3000}$?
A positive integer $N$ is called perfect if the sum of its divisors (including $1$ and $N$) is $2N$. A famous open problem is whether there is an odd perfect number.
Can someone confirm the following ...
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Liouville Lambda Function and Riemann Hypothesis
What is the exact statement involving the Liouville Lambda function, which is equivalent to Riemann Hypothesis, and true iff RH is true? Can anyone cite the sources for it and/or outline its proof in ...
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Confirmation of Equivalent Form of Riemann Hypothesis
Can anyone, who has knowledge of the following, share some more details about it because not much information is available publicly regarding the same:
RH is equivalent to the assertion that for all $...
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Minimum number of edges for a tree that joins the $27$ nodes of a $3 \times 3 \times 3$ regular grid
In 2014, Dumitrescu and Tóth (see Covering Grids by Trees, Figure 2) proved the existence of an inside-the-box tree consisting of $13$ connected line segments covering all the $27$ nodes of the ...
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Does the idea of creating a perfectly random problem to solve this have any merit, or is it completely useless quackery? [closed]
Statement- If perfect random proves P cannot equal NP
Explanation- The crux of P = NP is not figuring out the answer, but rather proving it, and the mathematical community has been approaching this ...
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Is any power of a theta function over primes a modular form?
Apologies for the vague title but I wasn't sure how to word this. To preface my question, let's recall the theta function:
$$\theta(\tau) = \sum_{n \in \mathbb Z} e^{i \pi n^2 \tau}$$
This function is ...
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Open problems in Lie symmetry theory
What are some famous open problems and conjectures in Lie symmetry theory? Is there some kind of list of these problems available or possibly a historical survey of the development of theory of ...
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Galois theory in combinatorics
When trying to find an explicit formula, how often shall we admit such a formula may not exist? To be more precise, suppose we are trying to find an explicit formula of a function $f(n)$ that returns ...
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Does My Conjecture on Selecting 'Special Nodes' in TSP Matrices to Eliminate 97-99% of Edges Hold Potential for Polynomial Time Solutions? [closed]
I was wondering something, let's say in a symmetric distance matrix of a sample of TSP, there was a sure algorithm that could remove around 97% of the values (weights or distances) that wouldn't ...
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Can we use the proof of the weak Goldbach conjecture to also prove the strong Goldbach conjecture?
Why doesn't proof of the weak Goldbach conjecture also prove the strong Goldbach conjecture?
Actually I am referring to this link. My question is why the logic used in this question cannot be used ...
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Is the weak Goldbach conjecture proved? [duplicate]
The Wikipedia page of the Goldbach's weak conjecture states that "In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture. As of 2018, the proof is widely accepted in the ...
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Is there an algorithm for this variant of the dominating set problem?
I stumbled upon this interesting variant of the dominating set problem lately, and as I have not been able to find a consecrated name, I suppose it has not been thoroughly studied yet.
The formulation ...
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A Number-theoretic Generalization of the Union-closed Sets Conjecture
Denote $\mathbb{N}^*=\{1,2,3,...\}, \dot k = \{k,2k,3k,...\}, \mathbb{P}=\{2,3,5,7,11,..\}$ and write $M\leq \mathbb{N}^*$ to denote that $M$ is lcm-closed, i.e. $a,b\in M\Rightarrow \text{lcm}(a,b)\...
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Open knight's tours on $n \times n \times \cdots \times n \subseteq \mathbb{Z}^k$ boards ($k \in \mathbb{N}-\{0,1\}$)
I would like to know under what conditions there exisits a (possibly open) knight's tour on a generic (hyper)cubic lattice $\{\{0,1,\ldots,n-1\} \times \{0,1,\ldots,n-1\} \times \cdots \times \{0,1,\...
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Research monographs and open problems in universal algebra
I am someone who is very interested in the mathematical subfield of universal algebra. I want to know, what are some significant open problems in universal algebra? I would like a list of such ...
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Has this weak version of Erdős Conjecture on arithmetic progressions been proven, or is it still an open problem?
This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker version of Erdős Conjecture, but I do not know how to prove it.
Erdős conjecture on arithmetic progressions ...
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Five new results on Conway's 99-graph problem [closed]
I realize that this editing won't make the question open (since this is against the guidelines to share results and ask to check them). Meanwhile, I'd like to replace a lot of text with a link so ...
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Is some twin prime average the sum of two twin prime averages, two ways?
Accoring to this question and a linked duplicate, it's been verified empirically up to some number that all twin prime averages greater than six, are the sum of two smaller twin prime averages.
I was ...
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Ruzsa–Szemerédi problem for regular graphs
The Ruzsa–Szemerédi problem asks for the maximum number of edges in a locally linear graph, i. e. a graph in which every edge belongs to a unique triangle (equivalently, any two adjacent vertices have ...
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Theory about NAND gate decompose
Recently, I play a game of Turing completeness where I utilize various gate circuits such as NAND, AND, and NOT to construct a circuit that satisfies the given truth table.
I didn't learn digital ...
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What are some conjectures of your own?
Background: Although this site is most-often used for specific one-off questions, many of the highest scored questions (also on MathOverflow), which gather a lot of attention to the site are about ...
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The set of all sums of 3 primes covers almost all of $2\Bbb{Z}+1$ (result of Helfgott), what can one say about $\Bbb{P} - \Bbb{P} = 2\Bbb{Z}$ problem?
Consider the set $\Bbb{P} = \pm$ the prime numbers and $\Bbb{P}_o$ is similarly $\pm$ the prime numbers other than $2$.
By Helfgott's result on the ternary Goldbach conjecture:
Every odd integer ...
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Does Han-Kobayashi attain the maximal rate-sum of a discrete memoryless very weak interference channel?
This figure describes an interference channel.
A discrete memoryless interference channel is said to be very weak if:
$$I(U_1;Y_1)\geq I(U_1;Y_2|X_2), ~~\forall ~ (U_1,X_1,X_2) \sim p(u_1,x_1)p(x_2),$...
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Marjorie Senechal (2011): "The question of packing tetrahedra is still unsolved" Is it still unsolved?
At about 15:49 in her 2011 talk Prof. Marjorie Senechal - "Quasicrystals Gifts to Mathematics":
But Hilbert understood that groups aren't everything and maybe not even the main thing. And ...
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Is equidistant points an open problem?
This post asks whether for any $n$-dimensional (presumably real) normed vector space, you can find $n+1$ equidistant points. They receive two answers saying that it is possible, but neither give much ...
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Every twin prime average $x \gt 6$ is the sum of two twin prime averages (Code checked up to $x \leq 1,000,000$).
If $p,q$ are a pair of twin primes, then $x = \dfrac{p+ q}{2} = q-1 = p+1$ is their twin prime average.
Conjecture. Every twin prime average $x \gt 6$ is the sum of two smaller twin prime averages, $...
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If $S = A\cdot A\cdot A$ generates $G$ and $f(S) = H$ is a group then $f(A) \leqslant H$ is also a group? Under what conditions is this true?
Let $G, H$ be two abelian groups and $f : G \to H$ a group homomorphism. Suppose that $G = \langle S \rangle$ and that $f(S) = H' \leqslant H$ a subgroup. In that case we say that $S$ represents $H'$...
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If $q^k n^2$ is an odd perfect number, then $n^2 - q^k = 2^r t$ implies that $3 \leq r$ is odd. Therefore?
The topic of odd perfect numbers likely needs no introduction.
Let $N$ be an odd perfect number given in the so-called Eulerian form $N = q^k n^2$ where $q$ is the special prime satisfying $q \equiv k ...
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Proving a variation of Lemoine's Conjecture by assuming the strong Goldbach Conjecture
In 2013, when I was just a totally newbie recreational mathematician, I read about Levy's conjecture (i.e., Lemoine's conjecture, stating that all odd integers greater than 5 can be represented as the ...
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Coman's Last Conjecture stating that every prime $q \geq 11$ can be written as $3 \cdot (p_1-1) + p_2$, where both $p_1$ and $p_2$ are prime numbers.
Today I was taking a look at Coman's book entitled Conjectures on Primes and Fermat Pseudoprimes, many based on Smarandache function (starting from the end, as I often do) and his last conjecture, the ...
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Sums of p-th powers of first N positive integers equal a p-th power of an integer
I am looking for $(p, N)$ where $p$, $N$ are integers greater than 1 and
$${\sum_{n=1}^{N}n^{p}}=M^{p}$$
where $M$ is an integer.
$p=2$, $N=24$ leading to $M=70$ is the Cannonball problem, and it was ...
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Summing the kth-nacci sequences over k
I've been playing around with an open problem I found in Peter Winkler's puzzle book. Roughly, it is
Let $C_p(n)$ be the expected length of the longest common subsequence of two random coin flip ...
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The same order type groups
Let $G$ be a finite group and $n$ be a natural number. Set $T(G)=\{g\in G|$ $%
g^{n}=1\}$ and $L_{n}(G)=|T(G)|$. Two finite groups $G_{1}$ and $G_{2}$ are
called of the same order type if and only if $...
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Are there infinitely many composite Euclid numbers of the second kind (Kummer numbers)?
$\displaystyle \prod_{i=1}^n p_i - 1$ is called Euclid number of the second kind (or Kummer number) , where $p_i$ is the i-th prime number.
It is not known whether there are infinitely many prime ...
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Another generalization of open mapping theorem
Let $T: E \to F$ be a linear continuous function between Banach spaces. Let
$$
B_E (x, 1) := \{z\in E \mid |z-x| <1\} \quad \text{and} \quad B_F (y, 1) := \{z\in F \mid |z-y| <1\} \quad \forall ...
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Name of conjecture about correlation of $\lambda(n)$ and $\lambda(n+1)$
I remember reading about a conjecture one night a while ago, but I can’t seem to find anything about it anymore. I have forgotten it’s name, but I believe the conjecture went as follows:
Suppose $\...
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Can we come up with a disjoint union of a subsets of the group $\Bbb{Z}$ such that they do not equal the cosets of a subgroup, yet they form a group?
If this applies to $\Bbb{Z}$ it probably will work for other groups $G$, however, for simplicity and because I'm interested in integers & their primes, let's work with $G = \Bbb{Z}$.
Anyway, we ...
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Doubt about approaching problem from the Kourovka notebook
Problem 20.58 from the Kourovka notebook, particularly the first part, reads
Is it true that for every positive integer $n$ there is a recognizable group that is the $n$-th direct power of a non-...
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Is there any other known relationship between even perfect numbers and odd perfect numbers, apart from their multiplicative forms?
(Note: This was cross-posted from MO, because it was not well-received there. Will delete the MO post in a few.)
Observe that an even perfect number $M = (2^p - 1)\cdot{2^{p - 1}}$ and an odd perfect ...
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Examples of well-known unsolved problems, in subjects don't related to number theory, that have several equivalent formulations
I know statements for some unsolved problems related in some way to number theory, being stated (I refer for each one of these unsolved problems) several equivalent formulations, for example: these ...
