3
votes
1answer
357 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 ...
0
votes
0answers
275 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 $\ ...
3
votes
2answers
396 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 ...
7
votes
4answers
1k views

Arbitrarily long arithmetic progressions

Are there arbitrarily long arithmetic progressions in which all the prime factors of all the terms are at most $N$, for some $N$? Assume all the terms are positive and the sequence of terms is ...
0
votes
0answers
190 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,\, ...
6
votes
2answers
464 views

On the prime number theorem in arithmetic progression

The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$ In a similar manner ...
13
votes
1answer
820 views

Small primes in arithmetic sequences

Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes $p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let ...
8
votes
1answer
451 views

On the least prime in arithmetic progressions

My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that $$p(a, q) \ll q^L$$ for ...
6
votes
2answers
1k views

least prime in a arithmetic progression

Hello Here I want to consider the simplest arithmetic progression $n\equiv 1\pmod{q}$ where $q$ is a prime. Is it true that we can find a prime $p\leq q^2$ in this arithmetic progression? This ...
5
votes
2answers
525 views

Asymptotic Distribution of Primes

Given an integer $n$ and let $1\leq m\leq n$ be such that $n$ and $m$ are coprimes define $$ \mathcal{N_{n,m}}:=\text{the set of primes $p$ such that $p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)$}. $$ ...
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 ...
7
votes
1answer
1k views

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 ...
4
votes
1answer
248 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 ...
1
vote
1answer
782 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. ...
6
votes
1answer
705 views

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 $x,q$? ...
16
votes
1answer
790 views

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. ...
16
votes
1answer
1k views

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 ...
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 ...
10
votes
1answer
2k views

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 ...
8
votes
2answers
490 views

Primes in quasi-arithmetic progressions?

Suppose $\alpha > 1$ is irrational. Are there infinitely many primes of the form $\left\lfloor \alpha n \right\rfloor$? Is the number of $p \leq X$ of this form $\sim \alpha^{-1} X ...
36
votes
1answer
8k views

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 $d$. ...
11
votes
2answers
522 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
1answer
997 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 ...
41
votes
3answers
6k views

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} ...
7
votes
1answer
601 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
2answers
408 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 ...
9
votes
1answer
554 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
4answers
1k views

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 ...