Questions tagged [open-problems]
If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"
590 questions
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No real roots of $\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}}$
Prove or disprove that $$\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}} \neq 0$$ for any real number $t$.
I find it surprising that so simple looking equations involving complex numbers ...
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Polynomial identification of natural numbers
Let $n$ be an integer parameter. Is there a polynomial $f(x,y,n)\in \mathbb{Z}[x,y,n]$ such that
$$
n\in \mathbb{N} \Longleftrightarrow \exists x,y\in \mathbb{Z}, f(x,y,n)=0?
$$
This is a generalized ...
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An inequality that implies Frankl's union-closed conjecture
Let $\mathcal{F}$ be an union-closed family of subsets of $[n]=\{1,2,...,n\}$, assume $\varnothing\in\mathcal{F}$. Let $l_i=|\{S|S\in\mathcal{F},i\notin S\}|,u_i=|\{S|S\in\mathcal{F},i\in S\}|$, then $...
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What are some open problems in geometric probability?
What are some open problems in geometric probability?
Context: My question, "A tetrahedron's vertices are random points on a sphere. What is the probability that the tetrahedron's four faces are ...
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Existence of n-Level Decomposition Trees for Large Primes
I am investigating a combinatorial structure on prime numbers called an n-Level Decomposition Tree (DT$_n$). The definition is as follows:
Let $p > 17$ be a prime number. An n-level decomposition ...
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Which pairs of mutually contradicting conjectures are there?
Years ago I had the pleasure of witnessing Simon Thomas giving a wonderful talk about Martin's conjecture, which I just now fondly remembered reading this question. Even though I am not well-versed in ...
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Is Landau's 4th problem the smallest unsolved problem in number theory?
The 4$^{th}$ of https://en.wikipedia.org/wiki/Landau%27s_problems is on the infinity of primes that are one more than a square.
The answer depends on certain assumptions about allowable statements and ...
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1
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An approach to a generalization of Frankl's union-closed sets conjecture
Let $I$ be a non-empty finite set, $\mathcal{F}$ be a non-empty union-closed family of subsets of $I$ except the empty set and $n$ real numbers $x_i\geq1,i\in I$. Let $d_i=\sum_{i\in J,J\in\mathcal{F}}...
- 1,558
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Pillai's arithmetical function and primality
Pillai's arithmetical function is defined by $P(n)=\sum_{k=1}^n\gcd(k,n)$.
Is it true that $1+P(n)\equiv 0\pmod{n}$ iff $n$ is prime?
It was stated without proof in 2014 on the OEIS A018804 and has ...
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Irrationally drifting Schwarz reflections
Let
$$ \alpha \in (0, \pi), \qquad \frac{\alpha}{\pi} \notin \mathbb{Q}, $$
and fix a bounded sequence of “drift-points”
$$ (c_k)_{k \in \mathbb{Z}} \subset \mathbb{C} $$
whose set of accumulation ...
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Tilted BBM Spines → Brownian Net?
Let $\mathcal{B}$ be critical binary branching Brownian motion on $\mathbb{R}$ started from a single particle at the origin. Fix $k \ge 1$ and a constant $\lambda > 0$. Condition on the atypical ...
3
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Can a uniformly-bounded Klein–Gordon wave sweep its nodal set across a rigid obstacle?
Consider Minkowski space $\mathbb R^{3+1}$ and the massive Klein–Gordon equation
$$ \square\varphi + m^{2}\varphi = 0 . $$
Let $\Omega\subset\mathbb R^{3}$ be a fixed smooth bounded “obstacle’’ with ...
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Furstenberg's topological proof method (of prime infinitude) might be more powerful than initially thought. $\sigma_0(\frac{p_n - 1}{4})$ odd i.o.?
Let $\sigma_0(n)$ be from number theory, i.e. the total number of divisors of an integer $n$. Let $p_n$ denote the $n$th prime number.
Let $S =$ the set of prime numbers $p \in 4\Bbb{Z} + 1$ such ...
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0
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More about the algebraic strengthening of Frankl's union-closed conjecture
Continue my previous question, consider the first conjecture:
Let $\mathcal{F}$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: ...
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A formula for minimum number of $k$-element nonnegative-sum subsets of a zero-sum set: Revisiting Manickam-Miklós-Singhi conjecture
Define the quantity $A(n,k)$ as follows:
$A(n,k)$ denotes the minimum number of $k$-element nonnegative-sum subsets of $n$ arbitrary real numbers $x_1,\dots,x_n$ whose sum is non-negative.
More ...
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