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First Hardy–Littlewood conjecture

Unanswered conjecture in number theory From Wikipedia, the free encyclopedia

First Hardy–Littlewood conjecture
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In number theory, the first Hardy–Littlewood conjecture[1] states the asymptotic formula for the number of prime k-tuples less than a given magnitude by generalizing the prime number theorem. It was first proposed by G. H. Hardy and John Edensor Littlewood in 1923.[2]

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Statement

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Let be positive even integers such that the numbers of the sequence do not form a complete residue class with respect to any prime and let denote the number of primes less than st. are all prime. Then[1][3]

where

is a product over odd primes and denotes the number of distinct residues of modulo .

The case and is related to the twin prime conjecture. Specifically if denotes the number of twin primes less than n then

where

is the twin prime constant.[3]

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Skewes' number

The Skewes' numbers for prime k-tuples are an extension of the definition of Skewes' number to prime k-tuples based on the first Hardy–Littlewood conjecture. The first prime p that violates the Hardy–Littlewood inequality for the k-tuple P, i.e., such that

(if such a prime exists) is the Skewes number for P.[3]

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Consequences

The conjecture has been shown to be inconsistent with the second Hardy–Littlewood conjecture.[4]

Generalizations

The Bateman–Horn conjecture generalizes the first Hardy–Littlewood conjecture to polynomials of degree higher than 1.[1]

Notes

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References

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