A pair of primes
that sum to an even integer are known as a Goldbach partition (Oliveira e Silva).
denote the number of Goldbach partitions of without regard to order, then the number of ways of writing
as a sum of two prime numbers taking the order of the two primes into account is
The Goldbach conjecture is then equivalent to the statement that or, equivalently, that , for every even integer .
A plot of ,
sometimes known as Goldbach's comet, for up to 2000 is illustrated above.
The following table summarizes the values of several variants of for , 4, ....