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3
votes
1answer
345 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
267 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
390 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
103 views

Arithmetic progression and 3^m,3^{m+1} intervals

I'm trying to prove (or disprove) the following "conjecture".Given the following set of powers of two: $$A = \{ x \mid x = 2^n \text{ and } 2^{n-1} < 3^m < x < 3^{m+1} < 2^{n+1}\}$$ ...
0
votes
1answer
107 views

Arithmetic progression and most significant digits in different bases

Given a number $x \geq 3$, let $b(x) \in \{0,1\}$ be the second most significant digit (bit) of its binary representation, and $t(x)\in \{1,2\}$ the most significant digit of its ternary ...
9
votes
1answer
501 views

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 ...
7
votes
2answers
409 views

Sum of divisor function over arithmetic progression

I am trying to find an estimate for the following sum: $$ \sum_{\substack{n \leq x \\ n \equiv k (m)}} d(n), $$ where $d(n)$ is number of divisors of $n$. I found estimates for the case when $k$ and ...
0
votes
0answers
189 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
453 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 ...
0
votes
0answers
163 views

Greedy sequences without k-term arithmetic progressions

If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k ...
3
votes
1answer
142 views

Non-asymptotically densest progression-free sets

For the context of this question, a progression-free set is a subset of integers that does not contain length-three arithmetic progressions. For large $N$, it is known that $[N] = \{1, \ldots, N\}$ ...
0
votes
1answer
363 views

Intersection of two arithmetic progressions

Using elementary matrix row and column operations on the system of two diophantine equations, namely, $N=an+b$ and $N=cn+d$, where $n\in\mathbb{N}^0$, it can be shown that the intersection of these ...
9
votes
2answers
365 views

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 ...
13
votes
1answer
816 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 ...
0
votes
2answers
202 views

sequences - recurrence relation [closed]

I have to find the expression of $(y_n)$ defined by : $$y_{n+1}=a y_n+b z_n+c$$ where $(z_n)$ is an arithmetico-geometric sequence : $$z_{n+1}=d z_n+e$$ and $a,b,c,d,e$ real numbers. Thank you ...
10
votes
0answers
398 views

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 ...
8
votes
1answer
756 views

Squares in an Arithmetic Progession

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 ...
8
votes
1answer
448 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 ...
12
votes
5answers
1k views

Special arithmetic progressions involving perfect squares

Some time ago the following rather easy problem appeared in an online publication called "Problems in Elementary NT" by Hojoo Lee: Prove that there are infinitely many positive integers $a$, $b$, $c$ ...
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 ...
2
votes
2answers
585 views

Sum of Series Where Exponent is Sum of Arithmetic Progression

Hi, How do i get the sum of such a sequence: $1 + x^{-1} + x^{-3} + x^{-6} + ...$ where the exponents are actually sum of arithmetic progression. i.e. $x^{-0} + x^{-(0 + 1)} + x^{-(0 + 1 + 2)} + ...
12
votes
2answers
995 views

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 ...
22
votes
1answer
925 views

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 ...
6
votes
1answer
431 views

Minimum cardinality of a difference set in $R^n$

Cross-posted from http://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn. Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the ...
10
votes
2answers
863 views

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 ...
2
votes
3answers
612 views

Arithmetic progressions of length 3 in subset of Z_n of size n^d

Let $A\subset\mathbb{Z}/n\mathbb{Z}$ such that: $|A|>n^{d}$ ($0< d <1$). Let $C=\{(x,y,2y-x)\in A\times A \times A\}$ be the set of $3$-term arithmetic progressions within $A$. [The ...
15
votes
2answers
856 views

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 ...
5
votes
2answers
524 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 $$ \mathcal{N_{n,m}}:=\text{the set of primes $p$ such that $p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)$}. $$ ...
5
votes
3answers
371 views

Structure of nonaveraging sets of integers

A set of integers is said to be nonaveraging if it contains no three-term arithmetic progression. I call a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ optimal when it has maximal ...
6
votes
1answer
291 views

Distribution of a function in an arithmetic progression

I am going to have to borrow the opening passage from Bombieri, Friedlander, Iwaniec${}^*$ since they state this idea so well. In the following $\|f\|$ means $\big(\sum_{n\leqslant x} ...
17
votes
4answers
1k views

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 ...
2
votes
0answers
921 views

Fun question in additive combinatorics

It is easy to see that for a finite set of integers $A$ of cardinality $n$, the cardinality of the sumset $A+A$ satisfies $$ 2n-1\leq |A+A|\leq \frac{n(n+1)}{2}. $$ The lower bound is essentially ...
6
votes
1answer
333 views

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, ...
0
votes
1answer
427 views

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
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 ...
4
votes
1answer
345 views

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 ...
12
votes
2answers
578 views

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 ...
5
votes
3answers
395 views

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 ...
1
vote
1answer
780 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
703 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 $x,q$? ...
0
votes
1answer
292 views

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 ...
5
votes
2answers
613 views

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 ...
15
votes
3answers
1k views

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 ...
16
votes
1answer
784 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. ...
7
votes
1answer
574 views

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 ...
11
votes
3answers
730 views

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, ...
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 ...