Strong Twin Prime Conjecture

Conjecture

It is conjectured that the number of twin primes less than or equal to $N \in \N$ is asymptotically equal to:

$2 C_2 \ds \int_2^N \dfrac {\d x} {\paren {\ln x}^2} = \dfrac {2 C_2 N} {\paren {\ln N}^2}$

The above is true only if the Twin Prime Conjecture holds.

Historical Note

The Strong Twin Prime Conjecture was proposed by Godfrey Harold Hardy and John Edensor Littlewood in $1923$, as a special case of the First Hardy-Littlewood Conjecture.

François Le Lionnais and Jean Brette present this as:

Un argument probabiliste montre que, s'il existe une infinité de nombres premiers jumeaux, alors de nombre de ceux qui sont situés dans l'intervalle $\sqbrk {x, x + a}$ est de l'ordre de $C \cdot \dfrac a {\paren {\Log x}^2}$ avec $C = 1,32 \ldots$

In English:

A probabilistic argument shows that, if there exists an infinite number of twin primes, then the number of those which are situated in the interval $\closedint x {x + a}$ is of the order of $C \cdot \dfrac a {\paren {\Log x}^2}$ where $C = 1,32 \ldots$

They also appear to attribute it to Viggo Brun.

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,32032 36316 \ldots$

• Weisstein, Eric W. "Twin Primes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TwinPrimes.html