# 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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### Minimum size integer accommodating some divisors within some prescribed gaps

Assume we pick $t$ uniformly random integers $l_1$ to $l_t$ independently from $1$ to $2^v$. Assume $k_1$ through $k_t$ are similarly picked from $1$ to $2^r$. What is the minimum size of non-...
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### Arithmetic progressions, given a prime

I have recently become interested in reading a little more on certain directions regarding primes in arithmetic progressions (AP). I would appreciate specific paper references (with the journal and ...
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### Prime-like numbers that avoid Green-Tao? [duplicate]

I would like to understand the conditions that support the Green-Tao Theorem, which established that the primes contain arbitrarily long arithmetic progressions. I am wondering: Q. Is it difficult ...
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### Primes in arithmetic progressions above a given threshold

Given co-prime $a,b$, Dirichlet's theorem states that there are infinitely many primes in the arithmetic progression $M = \{ a + bn : n \in \mathbb N\}$. Linnik's theorem asserts that the first such ...
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### Sufficient conditions on $a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often?

I am really interested in sufficient conditions on $a_i, b_i$ guaranteeing that the linear forms $a_1\phi(n)+b_1,\dots, a_k\phi(n)+b_k$ become simultaneously prime for infinitely many positive ...
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### Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern. We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
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### Explanation about arithmetico-geometric progression (AGP) [closed]

So I came across a formula that looks like: $x_n = \alpha x_{n-1} + \beta$ Since I don't have a strong mathematical background I didn't recognize it was an AGP and as I tried to express $x_n$ with ...
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### A reformulation of Erdős conjecture on arithmetic progressions

Erdős conjecture on arithmetic progressions states that if $S$ is a set of positive integers such that $c(S):=\sum_{n \in S} \frac{1}{n} = \infty$ (large set), then $\forall \ell \ge 3$ the set $S$ ...
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### Smallest set such that all arithmetic progression will always contain at least a number in a set

Let $S= \left\{ 1,2,3,...,100 \right\}$ be a set of positive integers from $1$ to $100$. Let $P$ be a subset of $S$ such that any arithmetic progression of length 10 consisting of numbers in $S$ will ...
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### Sums of two squares in arithmetic progressions

Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic ...
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### Covering integers by finitely many arithmetic progressions structure

Assume the positive integers $\mathbb{N}$ are partitioned as $$\mathbb{N} = \cup_{i = 1}^n (a_i + b_i \mathbb{N})$$ where $a_i, b_i \in \mathbb{N}$. Prove that all such partitions are obtained by the ...
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### Large finite subsets of Euclidean space with no isosceles (or approximately isosceles) triangles

Here's a question in combinatorial geometry which feels very much like other questions I'm familiar with but which I can't see how to get a hold of. I'll actually propose two different questions on ...
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### Partitioning the positive integers into finitely many arithmetic progressions

From Bóna's A Walk through Combinatorics: Prove or disprove that if we partition the positive integers into finitely many arithmetic progressions then there will be at least one arithmetic ...
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### Infinitely many primes in particular progressions

I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem? Let $d$ be a positive integer, $d\geq 2$. By Dirichlet's theorem, there is an ...
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### Homogeneous van der Waerden

The Erdős Discrepancy Problem is whether in any two-coloring of the naturals for any $C$ there is a sequence $d, 2d, \ldots nd$ such that the difference of red and blue numbers in it is more than $C$. ...
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### Extension of Dirichlet's Arithmetic Progression Theorem

Dirichlet's Arithmetic Progression Theorem states that: Given $a, b\in\mathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $k\in\mathbb{Z^+}.$ For any given $a$ and $b$ let ...
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### Arithmetic progression of rationals

We know that the set of rational numbers is countable. For which $n$ can we order all rational numbers as $a_1,a_2,\dots$ so that every subsequence of length $n$ is not an arithmetic progression? For ...
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### Product of arithmetic progressions

Let $(a_1,a_2\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,\dots,a_nb_n)$ is an ...
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### Does every prime $p$ appear in a $p$-term arithmetic progression of primes? [duplicate]

This is a follow-up to an earlier question. The answer to that question was found on this page. The discussion on OEIS seems to suggest that, for any prime $p$, there should exist a $p$-length ...
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### Is there an 11-term arithmetic progression of primes beginning with 11?

i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?
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### Closed set containing infinite arithmetic progressions of ANY gap

Let $A\subseteq [0,\infty)$ be a set containing infinite arithmetic progressions of ANY gap, that is, for any $d>0$, there is $t>0$ such that $t+kd\in A$ for all $k\in \mathbb N$. Molter and ...
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### Is a stronger version of the Erdős-Turan conjecture on arithmetic progessions reasonable? (And related questions.)

Define the size, possibly $\infty$, of a set $S\subseteq \mathbb{N}$ as $|S|=\sum\limits_{n\in S} \frac{1}{n}$. Then the Erdős-Turan conjecture states that if $|S|=\infty$, S must contain arbitrarily ...
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### large arithmetic progression modulo p (II)

Is it possible to construct a $B$ $\subseteq$ $Z_p(=Z/pZ)$ of cardinal $cp^{\frac{1}{3}}$, for some constant $c$, such that there exists an arithmetic progression of size $c_1p^{\frac{2}{3}}$, for ...
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### Generalized notion of divisor function?

Divisor function $d(n,m)$ counts the number of $q\in\Bbb N$ with $b<q<m$ such that $n\bmod q\equiv0$. Given $b>0$ what is the correct asymptotic, probabilistic and average case behavior of ...
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### Tighter upper bound for $\sum_{i=1}^kA_i\log(\frac{A_i}{e})$

What is the tightest upper bound one can obtain for the following expression $$\sum_{i=1}^kA_i\log(\frac{A_i}{e})$$ subject to $\sum_{i = 1}^k A_i = C$ in terms of $C$ and $k$? A very loose upper ...
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### Write {1,…,3n} as the disjoint union of arithmetic progressions of length 3 and steps 1, 2,…,n

For $n \equiv 0, 1, 2 \pmod 9$, write $\{\,1,\dots,3n\,\}$ as the disjoint union of arithmetic progressions $A_1, A_2,\dots,A_n$ of length 3, where $A_i$ has step $i$.
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### The original proof of Szemerédi's Theorem

Nowadays there are plenty of different proofs of the celebrated Szemerédi's Theorem but for historical reasons I would like to read and understand the original proof. The proof is very tricky and, for ...
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### Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...
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### Primes in arithmetic progression with a moduli equal to a power of 2

I am currently looking for a result stronger than Siegel-Walfisz theorem, which gives an upper bound on the error term $|\pi(x,a,b)-\frac{\pi(x)}{\phi(a)}|$ for particular $a$. The Siegel Walfisz is ...
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### Subsets of [1..N] with no three-term arithmetic progressions and no large gaps

Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow ...
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In contrast to classic results for arithmetic progressions of arbitrary length in one set at least of any finite partition of $\mathbb N$, it is easy to construct a partition in two sets of integers $... 1answer 247 views ### Primes in simultaneous arithmetic progressions Suppose we're given four positive integers$a$,$b$,$c$,$d$such that$a$and$b$are coprime, and$c$and$d$are coprime. Is there a non-negative integer$k$such that both$ak+b$and$ck+d$are ... 1answer 653 views ### Bounded gaps between primes in arithmetic progressions Has Zhang's work on bounded gaps between primes been extended to the following theorem? For any arithmetic progression$an+b,\gcd(a,b)=1$, there is a constant$H$(depending only on$a$) such that ... 1answer 394 views ### Siegel-Walfisz for the Möbius function I am working through the proof of the Bombieri-Vinogradov theorem in Analytic Number Theory (Iwaniec, Kowalski). My problem is that on page 424, it is said that$\mu(m)$satisfies$D_f(x;q,a)\ll (\...
Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where: 1) $s\in 2^{<\omega}$, 2) $N\in \mathbb{N}$, 3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid ...