# 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.

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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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### Primes in arithmetic progressions: weak version of Linnik's theorem with prime power modulus?

Looking at a problem in representation theory I ran into a question on small primes in arithmetic progressions. Let me begin with a short summary of results on small primes in arithmetic progressions. ...
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### Can we do better than random when constructing dense $k$-AP-free sets

We write $[N]$ to denote $\{1,\dots,N\}$. We say a set $S$ is $k$-AP-free if it lacks non-trivial arithmetic progressions of length $k$. We define the 2-color van der Waerden number, $w(2;k)$, to be ...
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### Primes in residue classes [duplicate]

For which sets of residue classes are there easy elementary proofs that there are infinitely many primes in them, which don’t require the machinery of proofs of Dirichlet’s theorem? Example: it’s ...
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### Remainder terms of congruence sums in sets of positive density

Let $\mathcal{A} \subset \mathbb{N}$ be an infinite sequence with positive density, in the sense that $$\tag{1} \lim_{x\to\infty} \frac{|\mathcal{A} \cap x|}{x} = c > 0,$$ and define the ...
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### Almost "dense" subsets of primes (and may be not only primes)

Based on Erdos idea, a subset of natural numbers is "dense" if the sum of its reciprocals is infinite, in such case it contains an infinite set of arithmetic progressions. Prime numbers are &...
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### What tools should I use for this problem?

Suppose we have $d$ cylindrical metal bars, with radius $l$, attached orthogonal to a support in random places: Now we have to attach bars with radius $k$ EVENLY SPACED, with distance $p$ between ...
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