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.

Sources

• 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.): $\text {II}$: The Series of Primes ($2$): $\S 2.7$. Further Results on formulae for primes: $\text {(iii)}$
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,3727 \ldots$