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First Hardy–Littlewood conjecture

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First Hardy–Littlewood conjecture
Plot showing the number of twin primes less than a given n. The first Hardy–Littlewood conjecture predicts there are infinitely many of these.
FieldNumber theory
Conjectured byG. H. Hardy
John Edensor Littlewood
Conjectured in1923
Open problemyes

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]

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]

Skewes' number

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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]

Consequences

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

Generalizations

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

Notes

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  1. ^ a b c Aletheia-Zomlefer, Fukshansky & Garcia 2020.
  2. ^ Hardy, G. H.; Littlewood, J. E. (1923). "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes". Acta Math. 44 (44): 1–70. doi:10.1007/BF02403921..
  3. ^ a b c Tóth 2019.
  4. ^ 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.

References

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