A prime partition of a positive integer 
is a multiset
of primes 
which sum to 
. For example, there are three prime partitions of 7 since
 |
The numbers of prime partitions of 
, 3, ... are 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12,
14, 17, 19, 23, 26, ... (OEIS A000607). If
for 
prime and 
for every other positive
integer 
,
then the Euler transform 
gives the number of partitions of 
into prime parts (Sloane and Plouffe 1995, p. 21).
The minimum number of primes needed to sum to 
, 3, ... are 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2,
1, 2, ... (OEIS A051034). The maximum number
of primes in a prime partition of 
is 
, where 
is the floor function.
For 
,
3, ..., these maxima are 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... (OEIS A004526),
corresponding to a representation in terms of all 2s for an even number or one 3
and the rest 2s for an odd number.
The numbers which can be represented by a single prime are obviously the primes themselves. Composite numbers which can be represented as the
sum of two primes are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ... (OEIS A051035), and composite numbers which are not
the sum of fewer than three primes are 27, 35, 51, 57, 65, 77, 87, 93, 95, 117, 119,
... (OEIS A025583). The Goldbach
conjecture would imply that no integer 
requires more than three primes.
The weak Goldbach conjecture, proved by
Helfgott (2014), implies the unconditional bound of four primes
for every integer 
.
