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)

for xy ≥ 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 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.[1] For example, an admissible k-tuple [2] (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.[3]


  1. ^ Richards, Ian (1974). "On the Incompatibility of Two Conjectures Concerning Primes". Bull. Amer. Math. Soc. 80: 419–438. doi:10.1090/S0002-9904-1974-13434-8. 
  2. ^ "Prime pages: k-tuple". Retrieved 2008-08-12. 
  3. ^ "447-tuple calculations". Retrieved 2008-08-12.