# 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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132 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 ...

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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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### 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} \...

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### 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}=\{\...

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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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### 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^...

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### 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 $...

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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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### 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 \...

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### 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 $...

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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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### 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)| &...

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### 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 $t$ ...

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### 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 $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...

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### 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 ...

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### Rational points on the unit circle

Is anything known about any of the following questions about rational points on the unit circle? By “double point” I mean an element of $2C$, where $C$ is the group of rational points on the unit ...

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### (Extremal) arithmetic combinatorics in non-abelian groups

Roth's Theorem states that any subset $A$ of $\{1, \dots, n\}$ with no solution to the equation $$x + y = 2z,\, (x, y, z) \in A^3,\, x \neq y$$ has size $o(n)$. Similar results hold when dealing with ...

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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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### Largeness and arithmetic progression properties of generic reals

Consider the following properties for a subset $A$ of $\mathbb{N}$:
(1) $A$ is large: $\sum_{n \in A}$$ 1\over n$$=\infty,$
(2) $A^\infty=\limsup \frac{|A \cap \{ 1, \dots, n\}|}{n} >0$,
(3) $A_\...

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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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### 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 $...

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### 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 ...

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### 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 ...

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### 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 (\...

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### Adding sets not containing arithmetic progressions of length three by forcing

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 $lh(s)$) $s$ ...

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### Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

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
...