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

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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.