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.
Assume there are fininitely many prime of that form .
Let .
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.