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We know that there is infinitely many of prime by the proof of Euclidean.and we know also that there are infinitely many of primes in arithmetic progression by the Dirichlet theorem

My question is, are there some special types of prime number that it has proven that there is infinitely many of them like the Fermat prime or the Sophie German prime or any other special types?

Abdo
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  • Does this general result count? https://math.stackexchange.com/questions/1250302/infiniteness-of-set-of-primes-such-f-have-root-mod-p – lhf Jun 15 '20 at 19:42
  • No.the special types that I am talking about have a relation with the integer $\mathbb{N}$ not a special polynomial or over some special field – Abdo Jun 15 '20 at 19:46
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    Does this count? There are infinitely many primes that divide Fibonacci numbers. – lhf Jun 15 '20 at 19:55
  • Yes can you give me a link or article tool about them .is this the only types that you know? – Abdo Jun 15 '20 at 19:57
  • It's conjectured that there are infinitely many repunit primes https://en.wikipedia.org/wiki/Repunit – Ethan Bolker Jun 15 '20 at 20:06
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    He's not interested in conjectured ones by the sounds of it. Also, see: https://arxiv.org/pdf/0905.1655.pdf , although I must admit, I don't know how useful it is in actually answering the question. But the title seems promising. – Adam Rubinson Jun 15 '20 at 20:50
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    https://en.wikipedia.org/wiki/Friedlander–Iwaniec_theorem is worth a look. (Sorry, for some reason the hyphen curtails the link. Copy and paste the entire url will work.) – Barry Cipra Jun 15 '20 at 23:17

2 Answers2

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  1. Famous Dirichlet theorem: Let $\ a\ $ and $\ b\ $ be relatively prime natural numbers. Then, there are infinitely many primes $\ p\equiv a\mod b.$

  2. Fascinating Friedlander–Iwaniec theorem: there are infinitely many primes $\ p=a^2+b^4\ $ where $\ a\ $ and $\ b\ $ are natural numbers.

Wlod AA
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There are infinitely many primes that divide Fibonacci numbers.

See for instance

and the original paper

lhf
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