Question

Prove that there are infinitely many primes of form \(8k-1\)


(quoting dirichlet thoerem is not allowed)

Answer 1

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

Answer 2

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.

Answer 3

@Callisto

Answer 4

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!

Answer 5

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

Answer 6

:)