Second Hardy–Littlewood conjecture

Template:Short description Template:Infobox mathematical statement In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was proposed by G. H. Hardy and John Edensor Littlewood in 1923.<ref name="original">Template:Cite journal.</ref>

Statement

The conjecture states that

<math>\pi(x+y) \leq \pi(x) + \pi(y)</math>

for integers Template:Math, where Template:Math denotes the prime-counting function, giving the number of prime numbers up to and including Template:Mvar.

Connection to the first Hardy–Littlewood conjecture

The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from Template:Math to Template:Math is always less than or equal to the number of primes from 1 to Template:Mvar. This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime Template:Mvar-tuples, and the first violation is expected to likely occur for very large values of Template:Mvar.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of Template:Math integers, while Template:Math. If the first Hardy–Littlewood conjecture holds, then the first such Template:Mvar-tuple is expected for Template:Mvar greater than Template:Math but less than Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

References

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External links

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