Is there an Infinite Number of Primes of Form n^2 + 1?

Open Question
Is there an infinite number of prime numbers of the form $n^2 + 1$?

Also known as
This problem is found according to some sources to be referred to as the Fifth Hardy-Littlewood Conjecture, but it is difficult to find hard evidence to back this up.

Landau's Problems
This is the $4$th of Landau's problems.