Prove that there are infinitely many primes of form $8k-1$

(quoting dirichlet thoerem is not allowed)

Question

Prove that there are infinitely many primes of form $8k-1$


(quoting dirichlet thoerem is not allowed)

amorfide · Answer

Suppose that $$p _{1},p_{2}...p _{l}$$ are the only prime numbers of the form 8k-1 for some integer k
it can be shown that
$$P=(p_{1}p_{2}...p_{l})^{2}-2 $$ is also of the form 8k-1 for some integer k

I think this is correct so far not sure how to continue

kainui · Answer

Failed but interesting attempt: $$\Large \text{let } k = 2^{p-3}$$ Then $$\Large 2^32^{p-3}-1 = 2^p-1$$ Then I had thought it was proven there was an infinite number of Mersenne primes, however it turns out that's just a conjecture so this proof doesn't work.

amorfide · Answer

@Callisto

rational · Answer

thats a good start @amorfide !

kai interesting way to pull mersenne into this  haha!
yeah if there are infinitely many mersenne primes then that also proves that there are infinitely many primes of form $2^nk-1$

that also proves that there are infinitely many even perfect numbers and many conjectures in number theory!

amorfide · Answer

http://mathhelpforum.com/number-theory/97357-primes.html maybe that will help, it doesn't actually complete the proof but it sort of explains it might be enough to get you going

ikram002p · Answer

:)