Twin Primes

Pairs of primes differing by 2: (3,5), (5,7), (11,13)... infinitely many? Conjecture still open.

0
Twin prime pairs up to N
0
Largest twin prime pair
0
Ratio to π(N)
1000
3
π₂(x) ~ 2C₂ · x/(ln x)² where C₂ = Π_{p≥3} p(p-2)/(p-1)² ≈ 0.6601

The twin prime constant C₂ ≈ 0.6601618... appears in the Hardy-Littlewood conjecture. Zhang (2013) proved there are infinitely many prime pairs with gap ≤ 70,000,000; the Polymath project reduced this to 246. If the Elliott-Halberstam conjecture holds, the bound would be 6.