Questions tagged [arithmetic-progression]
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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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} \}...
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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 $d$. ...
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A game on integers
$A$ and $B$ take turns to pick integers: $A$ picks one integer and then $B$ picks $k > 1$ integers ($k$ being fixed). A player cannot pick a number that his opponent has picked. If $A$ has $5$ ...
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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?
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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 ...
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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 ...
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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 ...
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Arithmetic Progressions of Squares
Fermat may or may not have known that there are 3-term arithmetic progressions of squares (like $1^2, 5^2, 7^2$, and that there are no 4-term APs. Murky history aside, Keith Conrad has two pleasant ...
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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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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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Arithmetic progressions inside polynomial sets
There are at most 3 perfect squares in arithmetic progression (Fermat, Euler). It was shown in [1] that if $n>2$ there are no three term arithmetic progression consisting of nth powers.
Take a non-...
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A proof of Van der Waerden's theorem using a weakened form of Szemeredi's theorem
Van der Waerden's theorem states that any colouring of the integers in a finite number of colours has monochromatic arithmetic progressions of arbitrary length. Szemerédi's Theorem is a dramatic ...
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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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Roth's theorem and Behrend's lower bound
Roth's theorem on 3-term arithmetic progressions (3AP) is concerned with the value of $r_3(N)$, which is defined as the cardinality of the largest subset of the integers between 1 and N with no non-...
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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 ...
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What tools should I use for this problem?
Suppose we have $d$ cylindrical metal bars, with radius $l$, attached orthogonal to a support in random places:
Now we have to attach bars with radius $k$ EVENLY SPACED, with distance $p$ between ...
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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 ...
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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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Arithmetic progressions in stopping time of Collatz sequences
Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.
We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
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Arithmetic progressions in power sequences
In connection with this MO post (and without any applications / motivation whatsoever), here is an apparently difficult - but nice - problem.
For a non-zero real number $s$, consider the infinite ...
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Special arithmetic progressions involving perfect squares
Prove that there are infinitely many positive integers $a$, $b$, $c$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $bc+1$, $ca+1$ are all perfect ...
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What are the analytic properties of Dirichlet Euler products restricted to arithmetic progressions?
There are (at least) two ways of writing down the Dirichlet L-function associated to a given character χ: as a Dirichlet series
$$\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$
or as an Euler product
$$\...
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Are there any papers about this observation of the distribution of the zeros of the zeta function?
Choose some $x > 1$. Then
$$
\lim_{T\to\infty} \sum_{\Im(\rho)<T}\cos(\ln(x)\Im(\rho))=-\infty
$$ where $\rho$ ranges over all zeros of the zeta function iff $x$ is prime or the power of some ...
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Small primes in arithmetic sequences
Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes
$p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let
$$Q_a(...
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A reformulation of Erdős conjecture on arithmetic progressions
Erdős conjecture on arithmetic progressions states that if $S$ is a set of positive integers such that $c(S):=\sum_{n \in S} \frac{1}{n} = \infty$ (large set), then $ \forall \ell \ge 3$ the set $S$ ...
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Smallest set such that all arithmetic progression will always contain at least a number in a set
Let $S= \left\{ 1,2,3,...,100 \right\}$ be a set of positive integers from $1$ to $100$. Let $P$ be a subset of $S$ such that any arithmetic progression of length 10 consisting of numbers in $S$ will ...
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Squares in an arithmetic progression
Let $P(x;a,b) := \{an+b, 0\leq n \leq x \} $ denote an arithmetic progression. Further let $A(x;a,b)$ denote the number of elements of $P(x;a,b)$ that are squares. It's an old conjecture of Rudin ...
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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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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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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 ...
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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(\...
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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
...
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What are the limits of the Erdős-Rankin method for covering intervals by arithmetic progressions?
To construct gaps between primes which are marginally larger than average, Erdős and Rankin covered an interval $[1,y]$ with arithmetic progressions with prime differences. A nice short exposition is ...
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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 ...
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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 ...
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What Dirichlet doesn't tell...
Let $n>1$ be an integer, and let us consider the set $P(n)$ of all prime numbers $p$ such that $p$ is not congruent to $1$ modulo $n$. Dirichlet's Density Theorem tells us that $P(n)$ has a natural ...
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4 squares almost in an arithmetic progression
Does there exist infinitely many coprime pairs of integers x,d such that x, x+d, x+2d, x+4d are all square numbers?
One example would be 49,169,289,529. This is the only example I have found so far ...
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A set with positive upper density whose difference set does not contain an infinite arithmetic progression
For $S \subset \mathbb{N}$ define $S-S=\{x-y:x \in S, y \in S\}$.
As noted below there is a simple example showing that a set $S \subset \mathbb{N}$ with positive upper density has a sumset $S+S=\{x+y:...
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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
...
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Sums of two squares in arithmetic progressions
Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic ...
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Bounded gaps between primes in arithmetic progressions
Has Zhang's work on bounded gaps between primes been extended to the following theorem?
For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that ...
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A weak form of the Erdős-Turán conjecture
This question is motivated by the answer of Gowers to the question Erdos Conjecture on arithmetic progressions.
Question. (1)-Suppose $A \subset \mathbb{N}$ is such that
Lim$_n$ $log(n) \cdot |A \...
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Largest number of k-arithmetic progressions without a (k+1)-arithmetic progression
Suppose $A \subseteq \{1,\dots,n\}$ does not contain any arithmetic progressions of length $k+1$. What is the largest number of $k$-term arithmetic progressions that $A$ can have? (one may also wish ...
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Are most primes in a prime arithmetic progression of length at least 3?
Following the following two previous questions on mathoverflow:
Are all primes in a PAP-3?
and
Covering the primes by 3-term APs ?
I have attempted to show that infinitely many primes are in an ...
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On the prime number theorem in arithmetic progression
The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$
In a similar manner ...
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Upper bound for number of k-term arithmetic progressions in the primes
Normal heuristics give that number of k-term arithmetic progressions in [1,N] should be about
$$c_k\frac{N^2}{\log^kN}$$
for some constant $c_k$ dependent on k. The paper of Green and Tao gives a ...
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Primes in quasi-arithmetic progressions?
Suppose $\alpha > 1$ is irrational. Are there infinitely many primes of the form $\left\lfloor \alpha n \right\rfloor$? Is the number of $p \leq X$ of this form $\sim \alpha^{-1} X (\log{X})^{-1}$...
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Largeness and arithmetic progression properties of generic reals
Consider the following properties for a subset $A$ of $\mathbb{N}$:
(1) $A$ is large: $\sum_{n \in A}$$ 1\over n$$=\infty,$
(2) $A^\infty=\limsup \frac{|A \cap \{ 1, \dots, n\}|}{n} >0$,
(3) $A_\...
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Product of arithmetic progressions
Let $(a_1,a_2\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,\dots,a_nb_n)$ is an ...
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Subsets of [1..N] with no three-term arithmetic progressions and no large gaps
Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow ...
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