One might wonder how it could be possible for an even number to exist that does not have such a Goldbach partition.
So, the sums cannot be distinct and there must be at least one even number between 4 and 14 with more than one Goldbach partition.
j] all of which evaluate to the same sum, that an even number somewhere is left without a Goldbach partition.
Now the only even prime, 2, belongs to a Goldbach partition in the case N = 4 but otherwise it cannot be part of a Goldbach partition since it would have to be paired with another even prime number.
Such numbers always have the Goldbach partition p + p.
But, while the probability that a particular totient sum will be a Goldbach partition might become very low, the increasing number of totients belonging to increasingly large even numbers ensures that the absolute expected number of Goldbach sums also grows without bound.
The Goldbach partitions of N, if there are any, are certainly contained in this initial set of partitions.
There exists a still smaller and more interesting subset of sums x + y that is guaranteed to contain all of the Goldbach partitions in most cases, the exception being the special case of even numbers that are twice a prime.
The totients thus can be formed into a set of sums t + t' = N and these sums must include most of the Goldbach partitions of N.
It has been noted (Australian Association of Mathematics Teachers [AAMT], 2016) that divisibility by 6 tends to confer on even numbers a relative richness in Goldbach partitions compared with numbers not divisible by 6.
Therefore, we expect, although we cannot insist on it, that numbers divisible by 6 will have about twice as many Goldbach partitions as the other types of even numbers.
These are not divisible by 6 and yet they have similar numbers of Goldbach partitions to their near neighbours that are multiples of 6.