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5
votes
3answers
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
votes
5answers
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
votes
1answer
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
1answer
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
votes
1answer
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 ...
4
votes
0answers
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
votes
2answers
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 ...
3
votes
1answer
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
1answer
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 ...
3
votes
2answers
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, ...
3
votes
1answer
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
1answer
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 $[N] = \{1, \ldots, N\}$ ...
3
votes
1answer
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 ...
2
votes
3answers
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 ...
2
votes
2answers
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
votes
1answer
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 ...
2
votes
0answers
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
1answer
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
vote
1answer
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. ...
1
vote
1answer
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
0
votes
1answer
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
votes
1answer
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 ...
0
votes
2answers
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 ...
0
votes
1answer
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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, ...
0
votes
0answers
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 $\ ...
0
votes
0answers
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: $$A = \{ x \mid x = 2^n \text{ and } 2^{n-1} < 3^m < x < 3^{m+1} < 2^{n+1}\}$$ ...
0
votes
0answers
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,\, ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...