The extended Goldbach conjecture, also called the Hardy-Littlewood extended Goldbach conjecture, gives a conjectured asymptotic formula
for the number of representations of a large even number
as the sum of two prime numbers.
Let 
count ordered pairs of primes 
such that 
.
Then, as 
tends to infinity through even values, the conjecture states that
 |
where 
is the twin primes constant and the product
is over the odd prime
divisors of 
(Hardy and Littlewood 1923, Halberstam and Richert 1974).
The conjecture predicts not merely that Goldbach partitions exist, but also their asymptotic abundance. In particular, it implies the Goldbach conjecture for every sufficiently large even number.
