All Questions
45 questions
1
vote
1
answer
325
views
Goldbach conjecture reformulation [closed]
As thought, the question below is a reformulation of the goldbach conjecture.
$ S = \{K - ap \mid a \geq 3, p \text{ is prime} < K/2 \} $, where $ a $ is an odd integer greater than or equal to 3, ...
- 23
4
votes
1
answer
251
views
Density of semiprimes in arithmetic progression
Let $n,a,b$ be integers such that $n$ and $a$ are coprime, and $n$ and $b$ are also coprime. According to the Prime number theorem for arithmetic progressions, the primes which are $a\mod n$ have the ...
- 654
2
votes
0
answers
125
views
Conditional stronger bounds on Linnik theorem with prime power modulus
This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.
By Linnik's ...
- 1,042
5
votes
1
answer
323
views
Primes in arithmetic progressions: weak version of Linnik's theorem with prime power modulus?
Looking at a problem in representation theory I ran into a question on small primes in arithmetic progressions.
Let me begin with a short summary of results on small primes in arithmetic progressions. ...
- 704
5
votes
1
answer
214
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Remainder terms of congruence sums in sets of positive density
Let $\mathcal{A} \subset \mathbb{N}$ be an infinite sequence with positive density, in the sense that
$$
\tag{1}
\lim_{x\to\infty} \frac{|\mathcal{A} \cap x|}{x} = c > 0,
$$
and define the ...
- 1,990
2
votes
1
answer
273
views
Primes in modular arithmetic progression
Fix a prime $p$.
I want to get $k<p$ primes $p_1<\dots<p_k$ such that at every $i\in\{1,\dots,k\}$ we have
$$p_i\equiv (2i+1+c)\bmod p$$ where $c$ is fixed and $2k+1+c<p$ holds.
For a ...
- 13.9k
1
vote
1
answer
177
views
Arithmetic progressions, given a prime
I have recently become interested in reading a little more on certain directions regarding primes in arithmetic progressions (AP). I would appreciate specific paper references (with the journal and ...
user137748
4
votes
1
answer
215
views
Primes in arithmetic progressions above a given threshold
Given co-prime $a,b$, Dirichlet's theorem states that there are infinitely many primes in the arithmetic progression $M = \{ a + bn : n \in \mathbb N\}$. Linnik's theorem asserts that the first such ...
- 183
3
votes
2
answers
417
views
Infinitely many primes in particular progressions
I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem?
Let $d$ be a positive integer, $d\geq 2$. By Dirichlet's theorem, there is an ...
- 33
5
votes
1
answer
416
views
Does every prime $p$ appear in a $p$-term arithmetic progression of primes? [duplicate]
This is a follow-up to an earlier question.
The answer to that question was found on this page. The discussion on OEIS seems to suggest that, for any prime $p$, there should exist a $p$-length ...
- 4,164
29
votes
4
answers
3k
views
Is there an 11-term arithmetic progression of primes beginning with 11?
i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?
- 4,164
26
votes
1
answer
1k
views
What is the status on this conjecture on arithmetic progressions of primes?
The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.
For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of ...
- 1,835
2
votes
2
answers
395
views
About consecutive integers covered by arithmetic progressions
Help me please to solve the following problem.
There are $n$ arithmetic progressions of the form:
$$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$
Initial integer terms $x_i \geq 0$ are varying.
...
- 267
3
votes
1
answer
359
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 ...
- 483
9
votes
1
answer
843
views
Bounded gaps between primes in arithmetic progressions
Has Zhang's work on bounded gaps between primes been extended to the following theorem?
For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that ...
- 28.2k
11
votes
2
answers
1k
views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
- 151k
4
votes
1
answer
980
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 ...
- 1,890
2
votes
0
answers
617
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 $\ \...
- 359
5
votes
2
answers
643
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
$$\alpha=0....
- 1,090
10
votes
4
answers
2k
views
Arbitrarily long arithmetic progressions
Are there arbitrarily long arithmetic progressions in which all the
prime factors of all the terms are at most $N$, for some $N$? Assume
all the terms are positive and the sequence of terms is ...
- 1,090
0
votes
0
answers
239
views
Conjecture about distribution of primes in arithmetic progression
For my work, i need the following
Conjecture: Let $N$ large number such that exist a prime number $q$ and $A>\frac{1}{2}$ such that $N^{1/2}<N^{A}\leq q-1<N.$ Then $\forall a\in\left[1,\, q\...
9
votes
2
answers
2k
views
On the prime number theorem in arithmetic progression
The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$
In a similar manner ...
- 319
9
votes
2
answers
1k
views
Are there five consecutive primes in arithmetic progression?
For example
3 consecutive primes in arithmetic progression
3,5,7 distance 2
151,157,163 distance 6
4 consecutive primes in arithmetic progression
...
14
votes
1
answer
1k
views
Small primes in arithmetic sequences
Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes
$p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let
$$Q_a(...
- 26k
8
votes
1
answer
863
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 some ...
- 463
11
votes
2
answers
3k
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 ...
- 2,508
1
vote
1
answer
214
views
Primes of the form $p_{i_1}p_{i_2}\cdots p_{i_n}+2k$
Let $S_{n,k}$ be the set of all numbers that can be written as the product of $n$ odd primes plus $2k$. Are there integers $n>1$ and $k>1$ such that $S_{n,k}$ contains finite number of primes?
- 61
5
votes
2
answers
641
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
$$
\mathcal{N_{n,m}}:=\text{the set of primes $p$ such that $p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)$}.
$$
...
- 3,626
4
votes
5
answers
2k
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 ...
8
votes
1
answer
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 ...
- 26.9k
5
votes
1
answer
310
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 ...
- 109k
2
votes
1
answer
1k
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. ...
7
votes
1
answer
833
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 $x,q$?
- 661
22
votes
1
answer
2k
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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 ...
- 9,114
4
votes
1
answer
3k
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 are ...
- 479
15
votes
1
answer
4k
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 ...
- 25.7k
8
votes
2
answers
576
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 $\sim \alpha^{-1} X (\log{X})^{-1}$...
- 13.1k
62
votes
1
answer
14k
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 $d$. ...
- 3,699
12
votes
3
answers
929
views
Mertens-like sum in arithmetic progressions
I find myself needing a good estimate for $\sum_{p\le x,\, p\equiv a\bmod q} 1/p$, perhaps something like
$$
\sum_{p\le x,\, p\equiv a\bmod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + O\big(\...
- 12.8k
3
votes
1
answer
1k
views
Strengthening of Dirichlet's theorem on arithmetic progressions
Hello all, Dirichlet's famous theorem asserts that any arithmetic progression
$\lbrace ax+b | x \in {\mathbb N}\rbrace$ contains infinitely many primes if a
and b are relatively prime.
I am ...
- 3,595
69
votes
4
answers
14k
views
Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?
It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} \}...
- 118k
8
votes
1
answer
880
views
Upper bound for number of k-term arithmetic progressions in the primes
Normal heuristics give that number of k-term arithmetic progressions in [1,N] should be about
$$c_k\frac{N^2}{\log^kN}$$
for some constant $c_k$ dependent on k. The paper of Green and Tao gives a ...
- 7,013
7
votes
2
answers
564
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 ...
- 13k
12
votes
1
answer
814
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 ...
- 2,901
16
votes
4
answers
2k
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 ...
- 156k