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]
Let m 1 , m 2 , … … --> , m k {\displaystyle m_{1},m_{2},\ldots ,m_{k}} be positive even integers such that the numbers of the sequence P = ( p , p + m 1 , p + m 2 , … … --> , p + m k ) {\displaystyle P=(p,p+m_{1},p+m_{2},\ldots ,p+m_{k})} do not form a complete residue class with respect to any prime and let π π --> P ( n ) {\displaystyle \pi _{P}(n)} denote the number of primes p {\displaystyle p} less than n {\displaystyle n} st. p + m 1 , p + m 2 , … … --> , p + m k {\displaystyle p+m_{1},p+m_{2},\ldots ,p+m_{k}} are all prime. Then[1][3]
where
is a product over odd primes and w ( q ; m 1 , m 2 , … … --> , m k ) {\displaystyle w(q;m_{1},m_{2},\ldots ,m_{k})} denotes the number of distinct residues of 0 , m 1 , m 2 , … … --> , m k {\displaystyle 0,m_{1},m_{2},\ldots ,m_{k}} modulo q {\displaystyle q} .
The case k = 1 {\displaystyle k=1} and m 1 = 2 {\displaystyle m_{1}=2} is related to the twin prime conjecture. Specifically if π π --> 2 ( n ) {\displaystyle \pi _{2}(n)} denotes the number of twin primes less than n then
is the twin prime constant.[3]
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]
The conjecture has been shown to be inconsistent with the second Hardy–Littlewood conjecture.[4]
The Bateman–Horn conjecture generalizes the first Hardy–Littlewood conjecture to polynomials of degree higher than 1.[1]
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