# Tagged Questions

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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### Bounds on the size of sets not containing a given finite pattern

Recall the following version of Szemerédi's Theorem: let $r_k(N)$ be the largest cardinality of a subset of $[N]:=\{1,\ldots, N\}$ which does not contain an arithmetic progression of length $k$. Then, ...
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### 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
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### 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 ...
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### Are most primes in a prime arithmetic progression of length at least 3?

Following the following two previous questions on mathoverflow: Are all primes in a PAP-3? and Covering the primes by 3-term APs ? I have attempted to show that infinitely many primes are in an ...
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### 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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### 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 ...
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### Arithmetic progressions modulo $p$ under the squaring map

I feel that the following problem should be known, but I'm not sure where to look for it. Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A_p$ denote the set of ...
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### 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 ...
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### 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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### Primes in arithmetic progressions

Denote by $\pi(x,a,q)$ the number of primes $p\le x$ of the form $p=qk+a$ and $E(x,a,q)=\phi(q)^{-1}\mathrm{Li}(x)-\pi(x,a,q)$. What is the strongest conjectured bound on $E(x,a,q)$ in terms of $x,q$?
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### 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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### Inverse Length 3 Arithmetic Progression Problem for sets with positive upper density

It is a famous theorem of Roth, which Szemerédi famously generalized, that if a set of natural numbers has positive upper density then it contains arithmetic progressions of length $k$. The famous ...
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### 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 ...
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### Covering the primes by arithmetic progressions

Define the length of a set of arithmetic progressions of natural numbers $A=\lbrace A_1, A_2, \ldots \rbrace$ to be $\min_i | A_i |$: the length of the shortest sequence among all the progressions. ...
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### Homogeneous arithmetic progressions in difference sets

I have a nasty feeling that I ought to be able to answer this question, but I've got other things to think about right now and I'm interested in the answer just so that I can reply to a mathematical ...
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### What does the computer suggest about the parity of p(n), for n in a fixed arithmetic progression?

Let p(n) be the number of partitions of n. A famous theorem of Euler allows one to compute the parity of p(n) quickly for quite large n. In: On the distribution of parity in the partition function, ...
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### Are all primes in a PAP-3?

Van der Corput [1] proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green & Tao [2] famously extended this theorem to length $k$.) But taking this in ...
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### 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 are ...
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### The Green-Tao theorem and positive binary quadratic forms

Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a ...
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### 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 ...
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### 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 ...
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### k-pseudorandom measures

In reading the paper of Green and Tao on arithmetic progressions within the primes, I became very interested in the notion of a k-pseudorandom measure discussed in that paper. A measure here is a ...
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### “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, ...