Number of Primes of Form n^2 + 1

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


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


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

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.