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.[2]
Statement
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
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.
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.
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.
Notes
References