This question already has an answer here:
We know that there are infinite number of primes so as there are infinite number of primes of the form $4n+3$ where $n\in Z^+$.
A note on Burton's book (Elementary Number Theory) somehow says that it is of high chance to expect that there are also infinite number of primes of the form $4n+1$. However it is not yet proven by the time the book was published.
A quick search on net gives infinitude of primes of the form $3n+1$ and $5n+1$.
The question is: The problem: Are there infinitely many number of primes of the form $4n+1$ still an open problem or it was already proven? If so, where can I find the proof? Thanks a lot.