# Tagged Questions

279 views

### Covering a finite subset of $\mathbb{N}$ with prime arithmetic progressions

Because of a problem I ran into I am trying to get a quick start in covering with arithmetic progressions. First I want to say I am aware of this previously asked question: Covering $\mathbb{N}$ with ...
358 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 ...
276 views

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 ...
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 ...
412 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 ...
760 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 ...
452 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 ...
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 $a$, $b$, $c$ ...
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 ...
999 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 ...
930 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 ...
864 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 ...
862 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 ...
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 $$\mathcal{N_{n,m}}:=\text{the set of primes p such that p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)}.$$ ...
292 views

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 $d$. ...
2k views

### 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 ...
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) + ...
677 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?.) Erdös in The Representation of a Graph by Set Intersections (1966) states: Theorem. Let $G$ be an arbitrary graph. Then ...
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 ...