Tagged Questions

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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About consecutive integers covered by arithmetic progressions

Help me please to solve the following problem. There are $n$ arithmetic progressions of the form: $$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$ Initial integer terms $x_i \geq 0$ are varying. ...
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Thin sets that are well-distributed over arithmetic progressions?

The primes do a nice job of intersecting an arithmetic progression $\{a+dn\}_{n=0}^\infty$ when $a$ and $d$ are coprime (see Dirichlet's theorem). I would like a set of integers $S$ such that the ...
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Covering a finite subset of $\mathbb{N}$ with prime arithmetic progressions

Because of a problem I ran into I am trying to get a quick start in covering with arithmetic progressions. First I want to say I am aware of this previously asked question: Covering $\mathbb{N}$ with ...
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Smallest prime in an arithmetic progression

Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...
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Arithmetic progressions in power sequences

In connection with this MO post (and without any applications / motivation whatsoever), here is an apparently difficult - but nice - problem. For a non-zero real number $s$, consider the infinite ...
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Arithmetic Progressions of Squares

Fermat may or may not have known that there are 3-term arithmetic progressions of squares (like $1^2, 5^2, 7^2$, and that there are no 4-term APs. Murky history aside, Keith Conrad has two pleasant ...
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Minimum cardinality of a difference set in $R^n$

Cross-posted from http://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn. Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the ...
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What Dirichlet doesn't tell…

Let $n>1$ be an integer, and let us consider the set $P(n)$ of all prime numbers $p$ such that $p$ is not congruent to $1$ modulo $n$. Dirichlet's Density Theorem tells us that $P(n)$ has a natural ...
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Arithmetic progressions of length 3 in subset of Z_n of size n^d

Let $A\subset\mathbb{Z}/n\mathbb{Z}$ such that: $|A|>n^{d}$ ($0< d <1$). Let $C=\{(x,y,2y-x)\in A\times A \times A\}$ be the set of $3$-term arithmetic progressions within $A$. [The ...
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Roth's theorem and Behrend's lower bound

Roth's theorem on 3-term arithmetic progressions (3AP) is concerned with the value of $r_3(N)$, which is defined as the cardinality of the largest subset of the integers between 1 and N with no non-...
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Asymptotic Distribution of Primes

Given an integer $n$ and let $1\leq m\leq n$ be such that $n$ and $m$ are coprimes define $$\mathcal{N_{n,m}}:=\text{the set of primes p such that p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)}.$$ ...
A set of integers is said to be nonaveraging if it contains no three-term arithmetic progression. I call a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ optimal when it has maximal ...