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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4 votes
2 answers
253 views

Binary words that are nonconstant on long arithmetic progressions

Let $w=x_0 x_1 x_2 \ldots$ be an infinite word, where each $x_i\in \{0,1\}$. For each positive integer $k$ (thought of as the jump size of an arithmetic progression) and each residue $0\leq a \leq k-...
13 votes
1 answer
1k views

Are there any papers about this observation of the distribution of the zeros of the zeta function?

Choose some $x > 1$. Then $$ \lim_{T\to\infty} \sum_{\Im(\rho)<T}\cos(\ln(x)\Im(\rho))=-\infty $$ where $\rho$ ranges over all zeros of the zeta function iff $x$ is prime or the power of some ...
2 votes
1 answer
265 views

Primes in modular arithmetic progression

Fix a prime $p$. I want to get $k<p$ primes $p_1<\dots<p_k$ such that at every $i\in\{1,\dots,k\}$ we have $$p_i\equiv (2i+1+c)\bmod p$$ where $c$ is fixed and $2k+1+c<p$ holds. For a ...
3 votes
0 answers
130 views

Does there always exist a monochromatic solution to ma+mb = nc+nd when m,n are coprime and N is coloured using 4 colours?

Let $m \ge 2 ,n \ge 2$ be positive integers which are coprime (that means that the greatest common divisor of $m,n$ is $1$). Is it possible to paint the set $\mathbf{N}$ of all natural numbers using $...
8 votes
1 answer
228 views

Arithmetic progressions in inverse image of totient function

I noticed on the OEIS that there are various sequences (e.g. A050515-A050520) that describe arithmetic progressions whose totients are all equal. For example, we have $$\varphi(\{1,2\}) = 1$$ $$\...
1 vote
1 answer
179 views

What are the hypotheses we should add for the generalizations of Furstenberg recurrence theorem?

In my question here I suggest a possibility for generalization of Furstenberg recurrence theorem needing some hypothesis for that generalization to be hold in the side of convergence of the below ...
23 votes
4 answers
3k views

What is the shortest route to Roth's theorem?

Roth first proved that any subset of the integers with positive density contains a three term arithmetic progression in 1953. Since then, many other proofs have emerged (I can think of eight off the ...
0 votes
0 answers
35 views

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-...
1 vote
1 answer
172 views

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 ...
4 votes
1 answer
211 views

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 ...
7 votes
1 answer
620 views

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 ...
-1 votes
1 answer
137 views

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 ...
13 votes
3 answers
1k views

What are the analytic properties of Dirichlet Euler products restricted to arithmetic progressions?

There are (at least) two ways of writing down the Dirichlet L-function associated to a given character χ: as a Dirichlet series $$\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ or as an Euler product $$\...
2 votes
0 answers
614 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ \...
27 votes
5 answers
8k views

Erdos Conjecture on arithmetic progressions

Introduction: Let A be a subset of the naturals such that $\sum_{n\in A}\frac{1}{n}=\infty$. The Erdos Conjecture states that A must have arithmetic progressions of arbitrary length. Question: I ...
2 votes
1 answer
256 views

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 ...
8 votes
1 answer
863 views

Upper bound for number of k-term arithmetic progressions in the primes

Normal heuristics give that number of k-term arithmetic progressions in [1,N] should be about $$c_k\frac{N^2}{\log^kN}$$ for some constant $c_k$ dependent on k. The paper of Green and Tao gives a ...
13 votes
2 answers
629 views

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 $S$ ...
12 votes
2 answers
719 views

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 ...
3 votes
1 answer
200 views

Gowers norms and three-term arithmetic progressions in the mean

Let $f:\mathbb{Z}^+\to \mathbb{C}$ be bounded. Say we are interested in studying how $f$ behaves in short three-term arithmetic progressions. It is very well-known that we can bound $$\sum_{h\leq H} \...
1 vote
0 answers
67 views

Wieferich primes and arithmetic prgressions

Let $p$ be an odd prime number. Let $K$ be a number field with Galois group $G$ and $H$ be a subgroup of $G$ stable under conjugation. Then the Cebotarev density theorem gives that $$\mathcal{L}=\{\...
3 votes
1 answer
398 views

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 ...
5 votes
0 answers
171 views

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 ...
2 votes
1 answer
426 views

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 ...
3 votes
2 answers
388 views

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 ...
26 votes
1 answer
1k views

What is the status on this conjecture on arithmetic progressions of primes?

The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes. For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of ...
2 votes
1 answer
292 views

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 $C$. ...
8 votes
2 answers
2k views

On the prime number theorem in arithmetic progression

The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$ In a similar manner ...
2 votes
1 answer
284 views

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 ...
1 vote
0 answers
146 views

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 ...
2 votes
0 answers
67 views

Discrepancy related independent vector from tensor product?

Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of the point set $P$ of $N$ points in $\mathbb Z^...
1 vote
0 answers
53 views

Discrepancy bound of integer tensor product sequence?

Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The (extreme) discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of the point set $P$ of $N$ points in $...
29 votes
4 answers
3k views

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?
5 votes
1 answer
409 views

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 ...
9 votes
1 answer
316 views

A weak form of the Erdős-Turán conjecture

This question is motivated by the answer of Gowers to the question Erdos Conjecture on arithmetic progressions. Question. (1)-Suppose $A \subset \mathbb{N}$ is such that Lim$_n$ $log(n) \cdot |A \...
1 vote
0 answers
58 views

Catch simple arithmetic progression with spiral bijection [closed]

Consider the simple arithmetic progression ($s, z \in \mathbb{Z}$): $a_1 = s$ $a_{n+1} = a_n + z = s + n\cdot z$ Can somebody devise a procedure (another progression) $b_n$ so that there exists a $...
2 votes
0 answers
70 views

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 ...
6 votes
0 answers
245 views

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 ...
1 vote
0 answers
96 views

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 ...
9 votes
2 answers
1k views

Are there five consecutive primes in arithmetic progression?

For example 3 consecutive primes in arithmetic progression 3,5,7 distance 2 151,157,163 distance 6 4 consecutive primes in arithmetic progression ...
1 vote
1 answer
229 views

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 ...
1 vote
1 answer
219 views

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 ...
6 votes
1 answer
407 views

A kind of anti-Ramsey result

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 $...
1 vote
0 answers
171 views

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 $i$.
12 votes
3 answers
892 views

Mertens-like sum in arithmetic progressions

I find myself needing a good estimate for $\sum_{p\le x,\, p\equiv a\bmod q} 1/p$, perhaps something like $$ \sum_{p\le x,\, p\equiv a\bmod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + O\big(\...
3 votes
0 answers
64 views

What's known about $X$ when $|X(n) + X(n)| < kn$, $n \in \mathbb{N}$, absolute constant $k$?

Let $X$ be an infinite sequence of integers$$x_1 < x_2 < x_3 < \ldots,$$and let $X(n)$ be the set$$\{x_1, x_2, \ldots, x_n\}.$$ Question. What is known about $X$ when we have$$|X(n) + X(n)| &...
2 votes
1 answer
421 views

Essential clarifications on application of pigeonhole principle

In here Lemma $4$ using pigeonhole says: For $T_1,\dots,T_s\in\Bbb R$ with $1\leq T_1,\dots,T_s<p$ and $\prod_{i=1}^sT_i > p^{s−1}$ and any integers $a_1,\dots,a_s$ there is an integer $t$ ...
-2 votes
1 answer
200 views

Solutions to a diophantine system

What is the smallest $\gamma_1,\gamma_2,\gamma_3>0$ such that given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ there are coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...
2 votes
1 answer
1k views

Covering Systems of infinite sets of residue classes mod primes

Take an infinite set of distinct primes and a (edit: or 2 , etc.) residue class for every prime. For exammple you can take all the primes bigger than some prime or the primes of a specific form (i.e. ...
2 votes
2 answers
381 views

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. ...