# Number of Primes of Form n^2 + 1

## Conjecture

Let $\map P x$ denote the number of prime numbers of the form $n^2 + 1$ less than $x$.

Then:

$\map P x \sim C \dfrac {\sqrt x} {\ln x}$

where:

 $\ds C$ $=$ $\ds \prod_{\substack {p \mathop > 2 \\ \text{p prime} } } \paren {1 - \dfrac {\paren {-1}^{\paren {p - 1} 2} } {p - 1} }$ $\ds$ $\approx$ $\ds 1 \cdotp 3727 \dotsc$

## Historical Note

According to François Le Lionnais and Jean Brette in their Les Nombres Remarquables of $1983$, this conjecture is the work of Godfrey Harold Hardy and John Edensor Littlewood, but the specific source of this information has not yet been tracked down.