The arithmetic-progression tag has no usage guidance.

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508 views

### Any rigorous way to claim that sums with repeat summands are few?

Let $B \subset \mathbb{Z}^+$. Define $r_{B,h}(n)$ to be the number of ways of writing $n$ as the sum of $h$ elements of $B$ and $R_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT ...

**4**

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**5**answers

1k views

### residue classes of primes, covering intervals and bounds on the different ways

Take the first $n$ primes $p_1,...,p_n$ and the primorial $P_n$ .Denote by $p_i$ every prime bigger than $p_n$ and smaller than $P_n$.
1) Is that true that there always be a number in any interval of ...

**4**

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**1**answer

215 views

### 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 ...

**4**

votes

**1**answer

362 views

### How large can a non-sumset be?

The theory of sumsets $A+B$ where $A$ and $B$ are finite subsets of an additive group $Z$ is extensively studied in additive combinatorics: finding long arithmetic progressions inside them, finding ...

**4**

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**1**answer

259 views

### non-asymptotic Bertrand-type theorems for arithmetic progression

It is well known that primes of form $4k+3$, call them $3=q_1 < q_2 < \dots$ satisfy $q_{n+1}/q_n\rightarrow 1$ (and even $q_n=\frac{n}{2\log n}(1+o(1))$). I would be glad to see results of ...

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**0**answers

208 views

### 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 ...

**3**

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**2**answers

455 views

### Primes from a Dirichlet sequence and an irrational number

From Dirichlet's theorem on arithmetic progressions, if $\text{gcd}(a,b)=1$ we know $\{ak+b\}_{k\ge 0}$ contains infinitely many primes. Let those primes be $p_1,p_2,\cdots$. Then the real
...

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**1**answer

757 views

### Can every finite graph be represented by an arithmetic sequence of natural numbers?

(This is a follow-up to my previous questions Natural models of graphs?.)
Erdös in The Representation of a Graph by Set Intersections (1966) states:
Theorem. Let $G$ be an arbitrary
graph. Then ...

**3**

votes

**1**answer

483 views

### 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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492 views

### “half arithmetic progressions” in dense sets

Fix a positive real number d>0. Szemeredi's theorem implies that for every integer k, there exists an integer N(k,d) such that if A is a subset of the interval [1,N] with density greater than d >0, ...

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**1**answer

1k views

### Strengthening of Dirichlet's theorem on arithmetic progressions

Hello all, Dirichlet's famous theorem asserts that any arithmetic progression
$\lbrace ax+b | x \in {\mathbb N}\rbrace$ contains infinitely many primes if a
and b are relatively prime.
I am ...

**3**

votes

**1**answer

159 views

### Non-asymptotically densest progression-free sets

For the context of this question, a progression-free set is a subset of integers that does not contain length-three arithmetic progressions.
For large $N$, it is known that $[N] = \{1, \ldots, N\}$ ...

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**1**answer

2k views

### Is there another proof for Dirichlet's theorem? [duplicate]

Possible Duplicate:
Is a “non-analytic” proof of Dirichlet’s theorem on primes known or possible?
Dirichlet's theorem on primes in arithmetic progression states that there ...

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**3**answers

627 views

### 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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**2**answers

791 views

### Sum of Series Where Exponent is Sum of Arithmetic Progression

Hi,
How do i get the sum of such a sequence:
$1 + x^{-1} + x^{-3} + x^{-6} + ...$
where the exponents are actually sum of arithmetic progression. i.e.
$x^{-0} + x^{-(0 + 1)} + x^{-(0 + 1 + 2)} + ...

**2**

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**1**answer

222 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 ...

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**0**answers

963 views

### Fun question in additive combinatorics

It is easy to see that for a finite set of integers $A$ of cardinality $n$, the cardinality of the sumset $A+A$ satisfies
$$
2n-1\leq |A+A|\leq \frac{n(n+1)}{2}.
$$
The lower bound is essentially ...

**1**

vote

**1**answer

144 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 ...

**1**

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**1**answer

798 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. ...

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**1**answer

313 views

### Do there exist sets of integers with arbitrarily large upper density which contains infinitely many elements that are not in an arithmetic progression of length 3?

Given that simply stipulating positive upper density is not sufficient to guarantee that all but finitely many members are in an arithmetic progression of length 3, that there indeed exists sets of ...

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42 views

### Arithmetic progressions on a graph

Given $K_k$ a complete graph what is the minimum $n\in\Bbb N$ needed so that there is a map:
$$f:\{0,1,\dots,n\}\rightarrow\mathsf{Edges}(K_k)$$ which makes every simple cycle to be $r$-term ...

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74 views

### 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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**1**answer

463 views

### greatest common divisor of p-1 and q-1 [closed]

Hi there,
Can we say that if $p$ and $q$ are distinct prime number diving $n$
$\Omega(gcd(p-1,q-1)) \leq \Omega(n)$
Where $\Omega(n)$ denotes the number of prime powers dividing $n$
Best
rahmi

**0**

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**1**answer

130 views

### Arithmetic progression and most significant digits in different bases

Given a number $x \geq 3$, let $b(x) \in \{0,1\}$ be the second most significant digit (bit) of its binary representation, and $t(x)\in \{1,2\}$ the most significant digit of its ternary ...

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295 views

### sequences - recurrence relation [closed]

I have to find the expression of $(y_n)$ defined by :
$$y_{n+1}=a y_n+b z_n+c$$
where $(z_n)$ is an arithmetico-geometric sequence :
$$z_{n+1}=d z_n+e$$
and $a,b,c,d,e$ real numbers.
Thank you ...

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**1**answer

332 views

### 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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44 views

### What is the maximum possible size of a $k$-average free subset of $[1, \dots, n]$?

Say that a subset $S \subset [1, \dots, n]$ is $k$-average free if, for all multi-subsets $A$ of $S$ with $|A| \le k$, the arithmetic mean of $A$ is not in $A$ (except for the trivial case when all ...

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150 views

### arithmetic progressions with few primes

Is this true ?
Let $\beta_0$ be a positive number. One may find $\beta>\beta_0$, $0<\lambda<1$, and infinitely many $q>1$ so that there exists an arithmetic progression of step $q$, $a_1, ...

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310 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 $\ ...

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116 views

### Arithmetic progression and 3^m,3^{m+1} intervals

I'm trying to prove (or disprove) the following "conjecture".Given the following set of powers of two:
$$A = \{ x \mid x = 2^n \text{ and } 2^{n-1} < 3^m < x < 3^{m+1} < 2^{n+1}\}$$
...

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210 views

### Conjecture about distribution of primes in arithmetic progression

For my work, i need the following
Conjecture: Let $N$ large number such that exist a prime number $q$ and $A>\frac{1}{2}$ such that $N^{1/2}<N^{A}\leq q-1<N.$ Then $\forall a\in\left[1,\, ...

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273 views

### Greedy sequences without k-term arithmetic progressions

If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k ...

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**1**answer

686 views

### Intersection of two arithmetic progressions

Using elementary matrix row and column operations on the system of two diophantine equations, namely, $N=an+b$ and $N=cn+d$, where $n\in\mathbb{N}^0$, it can be shown that the intersection of these ...