Questions tagged [open-problem]
Questions on problems that have yet to be completely solved by current mathematical methods.
391 questions
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On the Diophantine equation $x^{3}+y^{3}=(x+y)^{m}-(xy)^{n}$ — finiteness and partial results
I've been exploring some symmetric Diophantine forms that blend additive and multiplicative structures, and one equation I found keeps bugging me:
$$
x^3 + y^3 = (x + y)^m - (xy)^n,
$$
with integer $x,...
3
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Is there is a Jordan curve that approximates a circle, so that any “square inscribed in the curve” must be a tiny square and cannot be a large one?
With regards to the Inscribed Square problem, I would like to know if there is a Jordan curve that visually resembles a circle at a macroscopic scale, but at microscopic scales is extremely jagged, so ...
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Are $1,2,5,10$ the only solutions of $3^{n+1} \pmod {2^n} > 2^n-1.5^n$?
Introduction :
Put $r_n=2^n \{ 1.5^n \} $, i.e : the remainder of $3^n \pmod {2^n}$.
If the following inequality holds: $\forall n>1 , r_n<2^n-1.5^n$ (an open problem).
It has been proved that ...
- 560
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On the difference between consecutive maximal prime gaps
Let $g_n$ be the $n-$th prime gap. We say that $g_n$ is a maximal gap, if $g_m < g_n$ for all $m < n.$ Despite the fact that it is conjectured that prime gaps take every or most even numbers (...
- 25.2k
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Riemann-Liouville fractional integral operator norm and related singular value problem
For a while now I have been interested in the following (AFAIK) open problem: what is the $ L_p [0,1]$ norm of fractional integral operator $V^s: L_p[0,1] \to L_p[0,1] $ defined as
$$ (V^s f)(x) = \...
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An explicit closed-form formula to derive the subset of infinitely many odd “Collatz numbers”?
Remark : The current votes are $+15/-15$ .
I have a possible interesting question about the Collatz Conjecture, that I think it is "answerable".
Let me state my question as follows.
We are ...
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Is this "easier" Collatz conjecture (in particular, it is the Collatz conjecture with an additional rule) provable by elementary means?
https://en.wikipedia.org/wiki/Collatz_conjecture
Consider the same rules as the Collatz conjecture, but with the additional rule that if a number is divisible by $5,$ then we may choose to divide it ...
- 25.2k
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True/False: If $q\in\mathbb{Q}_{>1},$ then $\zeta(s)=q \implies s$ is irrational?
I don't know too much about the Riemann Zeta function other than its very well-known properties, but I am interested in the following.
Which value of $s\in\mathbb{R}_{>1}$ has the property that $\...
- 25.2k
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Are there specific main challenges in extending the inscribed-square theorem from smooth to continuous loops?
According to this video https://youtu.be/IQqtsm-bBRU?si=pSXEDQjTugHhhtI9 on Toeplitz' conjecture, if I've understood correctly, it is known that every smooth simple closed curve has an inscribed ...
- 236
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Is it known that all primes can be expressed as a square number minus a prime number?
i.e. Conjecture is that for every prime p, there exists an integer n such that $𝑝=𝑛^2−𝑞$ where q is prime.
e.g. $57593 = 240^2 - 7$
I assume it's either known / false / an entirely uninteresting ...
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Prove or disprove: $\forall a \ne k^2, \exists b, a^3 - b^2 \in \mathbb{P}$
In a forum post the following conjecture was proposed with no background information.
For all positive non-perfect-square number $a$, there exists integer $b$ such that
$$ a^3 - b^2 \text{ is a prime ...
- 1,072
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Can Dickson's conjecture with $b_i=1$ be proven for one $n,$ given that there are no obvious divisibility restrictions preventing this from happening?
I was reading this answer to it's question, and came across Dickson's Conjecture, because I was independently investigating the case where $b_i=1$ for $i\in\{1,2,\ldots,k\}.$
Dickson's conjecture says ...
- 25.2k
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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 ...
- 25.2k
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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 ...
- 41
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154
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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 $...
- 1,861
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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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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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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 ...
- 107
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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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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 ...
- 1,861
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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 ...
- 107
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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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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,\...
- 1,374
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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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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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