Twin prime constant $C_2$

edit · history · discussion · long url · probability theory number theory

Number

$C_2$

0.660161815846869573927812110014555778432623360284733413319448423335405642304495277143760031413839867911779

Definition

$C_2 := \prod_{p \geq 3} \big( 1 - (p-1)^{-2} \big)$, where the product ranges over the odd primes $p$.

Comments

(1)

Let $\pi_2(x)$ denote the number of primes $p\leq x$ such that $p+2$ is also a prime. The first Hardy-Littlewood conjecture implies that $\pi_2(x) \sim 2C_2\frac{x}{\log^2 x}$.

Links

[1]

OEIS: A005597

[2]

Wikipedia: First Hardy-Littlewood conjecture

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

How they were obtained: Checked here against mpmath's independent computation of the same constant, which agreed to past the hundredth digit. That is a stronger check than recomputing at two precisions, since the two computations share no code, but neither one bounds its own error.