Second Hardy–Littlewood conjecture

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 on prime k-tuples, but the first violation is likely to 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 x 10¹⁷⁴ but less than 2.2 x 10¹¹⁹⁸ .

References

• Engelsma, Thomas J. "k-tuple Permissible Patterns". Retrieved 2008-08-12.
• Hardy, G.H. (1923). "On some problems of "partitio numerorum" III: On the expression of a number as a sum of primes". Acta Math. 44: 1–70. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Oliveira e Silva, Tomás. "Admissible prime constellations". Retrieved 2008-08-12.
• Richards, Ian (1974). "On the Incompatibility of Two Conjectures Concerning Primes". Bull. Amer. Math. Soc. 80: 419–438.