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.

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,3727 \ldots$