Second Hardy–Littlewood conjecture
- π(x + y) ≤ π(x) + π(y)
for x, y ≥ 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. 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.
- Hensley, Douglas; Richards, Ian. "Primes in intervals". Acta Arith. 25 (1973/74): 375–391. MR 0396440.
- 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.
- "Prime pages: k-tuple". Retrieved 2008-08-12.
- "447-tuple calculations". Retrieved 2008-08-12.
- Engelsma, Thomas J. "k-tuple Permissible Patterns". Retrieved 2008-08-12.
- G. H. Hardy and J. E. Littlewood (1923). "On some problems of "partitio numerorum" III: On the expression of a number as a sum of primes". Acta Math. 44: 1–70. doi:10.1007/BF02403921.
- Oliveira e Silva, Tomás. "Admissible prime constellations". Retrieved 2008-08-12.
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