<Dd> lim inf n → ∞ (p n + 1 − p n) <N with N = 7 × 10 7, (\ displaystyle \ liminf _ (n \ to \ infty) (p_ (n + 1) - p_ (n)) <N \; (\ text (with)) \; N = 7 \ times 10 ^ (7),) </Dd> <P> is a major improvement on the Goldston--Graham--Pintz--Yıldırım result . The Polymath Project optimization of Zhang's bound and the work of Maynard has reduced the bound to N = 246 . </P> <P> The Hardy--Littlewood conjecture (named after G.H. Hardy and John Littlewood) is a generalization of the twin prime conjecture . It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem . Let π (x) denote the number of primes p ≤ x such that p + 2 is also prime . Define the twin prime constant C as </P> <Dl> <Dd> C 2 = ∏ p p r i m e p ≥ 3 (1 − 1 (p − 1) 2) ≈ 0.660161815846869573927812110014...(\ displaystyle C_ (2) = \ prod _ (\ textstyle (p \; (\ rm (prime)) \ atop p \ geq 3)) \ left (1 - (\ frac (1) ((p - 1) ^ (2))) \ right) \ approx 0.660161815846869573927812110014 \ dots) </Dd> </Dl>

How many twin primes are there from 51to 100