for integers x, y ≥ 2, where π(z) denotes the prime-counting function, giving the number of prime numbers up to and including z.

The statement of the second Hardy–Littlewood conjecture is equivalent to the statement 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 was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime k-tuples, and the first violation is expected to likely occur for very large values of x.

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.

[4] This number theory-related article is a stub.