Definition:Twin Primes Constant
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Definition
The twin primes constant is the real number:
\(\ds \Pi_2\) | \(:=\) | \(\ds \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} }\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 0 \cdotp 66016 \, 18\) |
This sequence is A005597 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Also defined as
François Le Lionnais and Jean Brette, in their Les Nombres Remarquables of $1983$, define the twin primes constant as:
\(\ds \Pi_2\) | \(:=\) | \(\ds 2 \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} }\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 1 \cdotp 32032 \, 36316 \ldots\) |
Also known as
Some sources (in particular François Le Lionnais and Jean Brette, in their Les Nombres Remarquables of $1983$, refer to this as the Shah-Wilson constant.
Research is required to identify who Shah and Wilson were, but the work they reported it in was published around $1919$ in Proceedings of the Cambridge Philosophical Society.
Some sources denote it $C_2$.
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,66016 18 \ldots$
- Weisstein, Eric W. "Twin Primes Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TwinPrimesConstant.html