Unanswered conjecture in number theory
In number theory, the first Hardy–Littlewood conjecture 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.
Statement.
Let formula_1 be positive even integers such that the numbers of the sequence formula_2 do not form a complete residue class with respect to any prime and let formula_3 denote the number of primes formula_4 less than formula_5 st. formula_6 are all prime. Then
formula_7
where
formula_8
is a product over odd primes and formula_9 denotes the number of distinct residues of formula_10 modulo formula_11.
The case formula_12 and formula_13 is related to the twin prime conjecture. Specifically if formula_14 denotes the number of twin primes less than "n" then
formula_15
where
formula_16
is the twin prime constant.
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
formula_17
(if such a prime exists) is the Skewes number for "P".
Consequences.
The conjecture has been shown to be inconsistent with the "second" Hardy–Littlewood conjecture.
Generalizations.
The Bateman–Horn conjecture generalizes the first Hardy–Littlewood conjecture to polynomials of degree higher than 1.
Notes.
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