Adapt the proof of Theorem 1.5.9 to prove:
Theorem 1.5.9 - There is an infinit number of primes.
(a) There is an infinite number of primes of the form 4n + 3.
(b) There is an infinite number of primes of the form 6n + 5.
Use Dirichlet's theorem on arithmetic progressions.
This website should tell you all you need to know
Dirichlet's theorem on arithmetic progressions - Wikipedia, the free encyclopedia
Using Dirichlet's theorem is complete overkill here.
This odd number can be factored into prime divisors.
It cannot be that each prime divisor has form because the product of numbers of the form still has the form . Therefore, there must be a prime divisor of the form . Thus, be divisible by one of . But that implies for . And this is impossible.
I think this is similar to the one above. Try modifing the proof.(b) There is an infinite number of primes of the form 6n + 5.