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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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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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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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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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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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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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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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) + ...
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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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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 ...
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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 ...