Second Hardy–Littlewood conjecture

In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. If π(x) is the number of primes up to and including x then the conjecture states that

π(x + y) ≤ π(x) + π(y)

for x, y ≥ 2.

This means that the number of primes from x + 1 to x + y is always less than or equal to the number of primes from 1 to y. This is probably false in general as it is inconsistent with the more likely first Hardy–Littlewood conjecture on prime k-tuples, but the first violation is likely to occur for very large values of x. For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of y = 3159 integers, while π(3159) = 446. If the first Hardy–Littlewood conjecture holds, then the first such k-tuple is expected for x greater than 1.5 × 10¹⁷⁴ but less than 2.2 × 10¹¹⁹⁸.