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### 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 ...
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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 $\ ... 1answer 820 views ### Small primes in arithmetic sequences Fix an integer$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$. Let ... 1answer 451 views ### On the least prime in arithmetic progressions My question concerns the least prime (denoted$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 ... 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 ... 1answer 430 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 1answer 2k 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 ... 2answers 704 views ### 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 ... 1answer 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$d$. ... 2answers 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) + ... 3answers 5k 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} ...
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 ...