A Goldbach partition of an even number 
is an unordered pair of primes 
such that 
(Oliveira e Silva). Letting 
denote the number of Goldbach partitions of 
without regard to order, the number of ways of writing 
as a sum of two prime
numbers taking the order of the two primes into
account, for 
, is
 |
(1)
|
The Goldbach conjecture is then equivalent to the statement that 
or, equivalently, that 
, for every even
integer 
.
The 
and 
convention above is used by Oliveira e Silva. The NIST Digital Library of Mathematical
Functions instead denotes by 
the number of unordered partitions into odd
primes for even integers 
. In analytic
number theory, Bhowmik and Halupczok (2020) use 
, where the sum is over primes,
for the ordered unweighted count. They use 
for the weighted count, where
is the Mangoldt function, while Brüdern
et al. (2019) denote this weighted function by 
. Veritasium (2025) uses 
for the unordered partition count. This 
is a local notation for Goldbach partitions and is unrelated
to the standard functions 
and 
that count prime factors.
For an even integer 
, each Goldbach partition determines a unique
larger prime number 
in the interval ![[N/2,N-2]](/images/equations/GoldbachPartition/Inline25.svg)
. Consequently,
 |
(2)
|
where 
is the prime counting function. Equality
holds iff 
is a prime number for every
prime number 
in that interval. Deshouillers
et al. (1993, p. 210) proved that 210 is the largest value of 
attaining this upper bound.
Here 
. Numberphile (2017) illustrates the equality
case.
A plot of 
, sometimes known as Goldbach's comet, for 
up to 2000 is illustrated above.
For an integer 
, the cumulative number
of Goldbach partitions is
![sum_(i=1)^nr(2i)=1+sum_(k=2)^(pi(n))[pi(2n-p_k)-(k-1)],](/images/equations/GoldbachPartition/NumberedEquation3.svg) |
(3)
|
where 
is the 
th
prime number (J. F. Martin, pers. comm., Dec. 4,
2010). The initial 1 counts 
. For each smaller odd
prime 
, the summand counts
the possible primes 
with 
, so every
remaining partition is counted exactly once.
The following table summarizes the values of several variants of 
for 
, 4, ....
| partition type | OEIS | values |
 1 or prime | A001031 | 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, ... |
prime | A045917 | 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ... |
odd prime | A002375 | 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ... |
Various fractal properties have been observed in Goldbach's partition (Liang et al. 2006).
." Math. Comput. 61, 931-934,
1993.Veritasium. "The Obviously True Theorem No One Can Prove."
Jun. 20, 2025.