Twin prime conjecture

The conjecture asserts that infinitely many prime numbers

p

have

p + 2

prime. The Hardy–Littlewood twin-prime asymptotic conjecture predicts their quantitative frequency and is stronger. The Selberg sieve supplies an upper bound of the predicted order but does not prove infinitude.

Hardy–Littlewood twin-prime asymptotic conjecture ( $# {p \leq N : p + 2 prime} \sim 2 C_{2} N / (lo g N)^{2}$ )

The local-density prediction for twin primes is the displayed asymptotic, with the twin-prime constant

C_{2}

. It is stronger than infinitude alone. The Selberg sieve proves the correct order of magnitude as an upper bound, but the parity problem in sieve theory prevents its elementary majorant from proving this conjectured asymptotic.

This positive convergent product corrects for the dependence of primality of

n

and

n + 2

at each odd prime. The associated singular series is

2 C_{2}

: the factor at two is two, and at an odd prime the local correction is

(1 - 2/ p) / (1 - 1/ p)^{2}

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Let's show them how it's done with primes + awk. Edit. They have a -d option which also shows gaps!!! Too strong:

sudo apt install bsdgames
primes -d 1 100 | awk '/\(2\)/{print $1 - 2, $1 }'

gives us the list of all twin primes up to 100: