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# 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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### Density of semiprimes in arithmetic progression

Let $n,a,b$ be integers such that $n$ and $a$ are coprime, and $n$ and $b$ are also coprime. According to the Prime number theorem for arithmetic progressions, the primes which are $a\mod n$ have the ...
323 views

### Does Szemerédi's theorem hold for sets with positive upper Banach density?

We say that a set of natural numbers $A\subseteq \omega$ has positive upper density if $$\lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1} > 0.$$ Szeméredi's theorem states that every $A\subseteq \omega$ ...
300 views

### Where odious numbers meet arithmetic progressions

Suggested by this problem: Do the sets of all odious / evel numbers meet every infinite arithmetic progression? A number is odious if it contains an odd number of digits $1$ in its binary ...
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### Are there infinitely long arithmetic progressions in every increasing sequence of positive integers with bounded gaps between consecutive terms?

Suppose the largest gap is D>1 and at least two of the gaps 1,2,...,D appear infinitely many times. I think the answer is NO. But I find it difficult to formulate a necessary and sufficient ...
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### Bounds for Szemerédi’s theorem for GAP’s

Let a $(k,D)$-AP refer to sets of the form $\{n_0+l_1n_1+\dotsb +l_Dn_D: l_1,\dotsc,l_D \in [k]\}$ with cardinality $k^D$ (i.e. a proper $D$-dimensional GAP with width $k$). Let $r(N,k,D)$ be the ...
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### Progressions in sumset or complement

Fix $\epsilon>0$. For all large $N$, does there exist $A\subset [N]:=\{1,\dots,N\}$ such that both $A+A$ and $A^c:=[N]\setminus A$ lack arithmetic progressions of length $N^\epsilon$? I am aware ...
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1 vote
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### 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 ...
1k views

### Is there any relationship between Szemerédi's theorem and Sunflower conjecture?

I have observed some similar things between a reformulation of the Sunflower conjecture (see also conjecture 1.3 in Improved bounds for the sunflower lemma) and Szemerédi's theorem such that for ...
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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-...
1 vote
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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 ... 603 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 ...
198 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 $S$ ...
686 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 ...
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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 ...
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 $$\sum_{h\leq H} \... 1 vote 0 answers 64 views ### Wieferich primes and arithmetic prgressions Let p be an odd prime number. Let K be a number field with Galois group G and H be a subgroup of G stable under conjugation. Then the Cebotarev density theorem gives that$$\mathcal{L}=\{\...
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 ...