Second Hardy–Littlewood conjecture

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In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. The conjecture states that

π(x + y) ≤ π(x) + π(y)

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

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][2] For example, an admissible k-tuple [3] (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.[4]


  1. ^ Hensley, Douglas; Richards, Ian. "Primes in intervals". Acta Arith. 25 (1973/74): 375–391. MR 0396440. 
  2. ^ 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. 
  3. ^ "Prime pages: k-tuple". Retrieved 2008-08-12. 
  4. ^ "447-tuple calculations". Retrieved 2008-08-12.