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 $\displaystyle p_1,...,p_n$.
Let $\displaystyle N = 4(p_1\cdot .... \cdot p_n) - 1$.
This odd number can be factored into prime divisors.
It cannot be that each prime divisor has form $\displaystyle 4k+1$ because the product of numbers of the form $\displaystyle 4k+1$ still has the form $\displaystyle 4k+1$. Therefore, there must be a prime divisor of the form $\displaystyle 4k+3$. Thus, $\displaystyle N$ be divisible by one of $\displaystyle p_1,...,p_n$. But that implies $\displaystyle p_i | 1$ for $\displaystyle 1\leq i \leq n$. 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.