Questions tagged [arithmetic-progression]
An arithmetic progression is a (possibly infinite) sequence of numbers such that the difference between consecutive terms is always the same value.
8 questions from the last 365 days
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Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?
For a single-sorted algebraic theory $\mathcal{T}$ denote by $t_n$ the number of $\mathcal{T}$-algebras with $n$ elements (up to isomorphism). Is there an example for $\mathcal{T}$ such that ...
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2
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Explicit construction of integers with prescribed digit sum and residue class conditions
Let $q\geq 2$ be an integer, and $p,m\in \mathbb{N}$. Let $S_q$ be the function sum of digits in base $q$. If $\gcd(q-1,m)=1$, I was wondering if there is simple way to construct $k\in \mathbb{N}$ ...
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2
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APs in sumsets of exponential growing sequences
I posted this initially on SE, but after I didn't found a particular reference on it, I decided it would be more appropriate to post it here. A friend shared this observation with me and I thought ...
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4
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Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$
I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
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Expected number of coin flips before you see a $k$-term arithmetic progression of heads
Let $\{X_i\}_{i \in \mathbb Z_+} $ be independent fair coin flips. Write $S := \{i \in \mathbb Z_+\, | \, X_i \text{ is heads}\}$, and define, for an integer $k \geq 3$,
$$Y := \inf \{n \in \mathbb N \...
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Infinite set intersection with arithmetic progressions
Let $\mathcal{A}$ be the set of all arithmetic progressions in $\mathbb{N}$ i.e
\begin{align*}
\mathcal{A} = \{a + b\mathbb{N} : a,b\in\mathbb{N}, b\neq 0\}.
\end{align*}
Does there exist a set $X \...
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Goldbach conjecture reformulation [closed]
As thought, the question below is a reformulation of the goldbach conjecture.
$ S = \{K - ap \mid a \geq 3, p \text{ is prime} < K/2 \} $, where $ a $ is an odd integer greater than or equal to 3, ...
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Does every big polyomino contain a big arithmetic progression?
Define a $k$-AP (arithmetic progression) as $k$ vertices whose $x$- and $y$-coordinates both from an arithmetic progression, for example, (1,0), (2,2), (3,4) is a 3-AP.
Is it true that for every $k$ ...
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