Questions tagged [arithmetic-progressions]
Questions related to arithmetic progressions, which are sequences of numbers such that the difference between consecutive terms is constant
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Is there a subset of the positive integers with this property?
This question was partly inspired by my previous one, here: A strengthening of the Green-Tao theorem. As in that question, I will restate the definition of a maximal arithmetic progression. Let $k$ be ...
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Use the sampling argument to prove density Szemeredi (supersaturation) problem
This Exercise comes from Graph Theory and Additive Combinatorics (Yufei Zhao). I have tried for several days but end up with no idea.
Exercise 1.3.7. (Density Szemeredi). Let $k\geq 3$. Assuming ...
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A strengthening of the Green-Tao theorem
The Green-Tao theorem states that there are arbitrarily large arithmetic progressions in the set of primes. I am interested in a strengthening of this theorem. First, a definition. Let $k$ be a ...
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Find $a_9$ in arithmetic sequence.
Let $\{a_n\}_{n\in\mathbb{N}}$ be an aritmetic sequence with difference $b, b\in\mathbb{N}$ and $S_n=\sum\limits_{i=1}^n a_n$ be partial sum of $\{a_n\}_{n\in\mathbb{N}}$.
Let $\{g_n\}_{n\in\mathbb{N}}...
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A version of Van der Waerden's theorem
Recall that Van der Waerden's theorem states: Whenever one partitions $\mathbb{N}=A_0\sqcup A_1$, there is $i\in\{0,1\}$ such that $A_i$ contains arithmetic progressions of arbitrary length.
Recently, ...
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Are there Salem-Spencer sets with bounded absolute second forward difference?
A Salem-Spencer set is a set of numbers no three of which form an arithmetic progression.
Suppose $A$ is a Salem-Spencer set. And let $(a_n)_{n=1}^{\infty}$ be the set $A$ written as a strictly ...
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Understanding a character calculation for arithmetic progressions of squares in finite fields
so the problem I'm interested in is to show that all sufficiently large finite fields contain an arithmetic progression of 9 distinct perfect squares.
A professor in my department had some previous ...
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"Relationship between row A.P. in a symmetric matrix and column A.P. in "
Matrix A.P. relationship:(P. Shiva Shankar)
I am P. Shiva Shankar, a high school student, and I recently discovered the following result. I kindly seek feedback and endorsement from the community.
Let ...
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Are bounded prime gaps robust to adversarial residue-class deletion?
Following up on my previous question, suppose we fix parameters $\theta \in (0, 1/2)$ and $\eta > 0$. For each large $X$, an adversary chooses a set $\mathcal{R}(X)$ of residue classes of the form $...
user1675092
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Maximum amount of 3-term arithmetic progressions from a set of integers.
For each $n\in\mathbb{N},$ what set $X:=\{x_1,x_2,\ldots,x_n\}\subset\mathbb{N},$ with $x_1<x_2<\ldots<x_n,$ maximises $\pi(X),$ the amount of pairs $(x_k,x_m)$ with $1\leq k<m\leq n,$ ...
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Simultaneous prime-free short intervals modulo small $q$
Fix a large parameter $X$. For $0<\delta<\theta<\tfrac12$, set $H := X^{\theta}$.
Question. Is it true that there exist infinitely many values of $X$ and, for each such $X$, a modulus $q\le X^...
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Single-AP two-form dispersion beyond $1/2$ on short-interval averages?
Fix an even integer $h \geq 12$ and set $W = \prod_{p < h} p$. Choose a residue class $b \pmod{W}$ with the covering property that for every $s$ in the range $1 \leq s \leq h-1$, there is a prime $...
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Are all coprime residue classes $\bmod q$ appearing in the set of divisors of the sequence $p^n - k$?
Let $p$ be a prime and $k$ a non-zero integer, with gcd$(p, k) = 1$. If $q$ is any integer and $r$ is a residue mod $q$, with gcd$(r, q) = 1$, does there exist some divisor $d$ of some element of the ...
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Do symmetric 3×3 matrices with rows in arithmetic progression always produce A.P. columns in matrix products? [closed]
P. Shiva Shankar — High school student
I recently observed an interesting and seemingly undocumented property of matrix multiplication:
Let A be a 3×3 symmetric matrix in which each row is an ...
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Progressions in the entrance exam.
Let $a, b, c \in \mathbb{R}$ be an arithmetic progression with common difference $r$, i.e.,
$$
b = a + r, \quad c = a + 2r.
$$
The numbers $a - 1$, $b$, and $c+4$ form a geometric progression with ...
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