Second Hardy-Littlewood conjecture

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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)

where 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, but the first violation is likely to occur for very large values of x. For example, an admissible k-tuple [1] (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 x 10¹⁷⁴ but less than 2.2 x 10¹¹⁹⁸ [2].