The arithmetic-progression tag has no wiki summary.

**34**

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### Is the Green-Tao theorem true for primes within a given arithmetic progression?

Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...

**17**

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**5**answers

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### Finitely many arithmetic progressions

A few years ago, somebody told me a lovely problem. I suspect there may be more to it (which I would be interested in learning), and would very much like to find a reference, it makes me uncomfortable ...

**11**

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**2**answers

500 views

### Mertens-like sum in arithmetic progressions

I find myself needing a good estiamate for $\sum_{p\le x,\, p\equiv a\mod q} 1/p$, perhaps something like
$$
\sum_{p\le x,\, p\equiv a\mod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + ...

**3**

votes

**1**answer

650 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?.)
ErdÃ¶s in The Representation of a Graph by Set Intersections (1966) states:
Theorem. Let $G$ be an arbitrary
graph. Then ...

**3**

votes

**1**answer

984 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 ...

**40**

votes

**3**answers

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### Is a “non-analytic” proof of Dirichlet's theorem on primes known or possible?

It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} ...

**21**

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**5**answers

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### Erdos Conjecture on arithmetic progressions

Introduction:
Let A be a subset of the naturals such that $\sum_{n\in A}\frac{1}{n}=\infty$. The Erdos Conjecture states that A must have arithmetic progressions of arbitrary length.
Question:
I ...

**13**

votes

**1**answer

582 views

### Goldbach-type theorems from dense models?

I'm not a number theorist, so apologies if this is trivial or obvious.
From what I understand of the results of Green-Tao-Ziegler on additive combinatorics in the primes, the main new technical tool ...

**7**

votes

**1**answer

589 views

### Upper bound for number of k-term arithmetic progressions in the primes

Normal heuristics give that number of k-term arithmetic progressions in [1,N] should be about
\[c_k\frac{N^2}{\log^kN}\]
for some constant $c_k$ dependent on k. ...

**5**

votes

**2**answers

390 views

### Smallest k-term AP of primes

Let $S(k)$ denote the smallest integer such that there exists a k-term arithmetic progression of primes among the integers $[1,S(k)]$. Green and Tao have an unpublished note that gives a very large ...

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**2**answers

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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 ...

**3**

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**2**answers

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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, ...

**9**

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**1**answer

539 views

### Covering the primes by 3-term APs ?

Hello, the Green-Tao theorem says infinitely many k-term Arithmetic Progressions exist for any integer k.
My question is: can we actually partition the primes into 3-term APs only (or is there a ...

**13**

votes

**4**answers

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### Arithmetic progressions without small primes

The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...