[UNSOLVED] Show there are infinitely many primes congruent to $\pm 2 \mod 5$ without using Dirichlet's Prime Number Theorem/Dirichlet's Theorem on Primes in Arithmetic Progressions. In particular, prove it in a style similar to Euclid's proof of infinite primes.

This was a fun extra credit problem on a test I had a couple months back.

PS. Please don't post a link to the answer.
PPS. If you don't know how Euclid proved there were infinite primes, follow jhonyy9's advice and look it up on wikipedia.

Question

[UNSOLVED] Show there are infinitely many primes congruent to $\pm 2 \mod 5$ without using Dirichlet's Prime Number Theorem/Dirichlet's Theorem on Primes in Arithmetic Progressions. In particular, prove it in a style similar to Euclid's proof of infinite primes.

This was a fun extra credit problem on a test I had a couple months back.

PS. Please don't post a link to the answer.
PPS. If you don't know how Euclid proved there were infinite primes, follow jhonyy9's advice and look it up on wikipedia.

jhonyy9 · Answer

- ohhh ! YES !!! - realy this is one very very nice problem of primes ...
- so if do you know right,good the Euclid's proof of infinite primes ,so then here you have not problems to make this proof the same like ...

Good luck !!!


bye 


- for help you can check this Euclid's proof on wikipedia.org -- there is understandably wrote sure ...

- thank you for this exersice ...

anonymous · Answer

:( The positions are reversed, KingGeorge, with you posting the hard question. But I can't answer this one. :P

kinggeorge · Answer

This is only hard because it relies on a somewhat clever trick, and it requires some knowledge of the properties of divisibility. Most of your problems just look downright impossible.