Is every sufficiently large even integer the sum of distinct primes? > Is every sufficiently large even integer the sum of (any number of) _distinct_ primes? No doubt this question has been asked before; does the conjecture/theorem have a name? It is related to Goldbach's conjecture, which states that every even integer is the sum of _two_ (not necessarily distinct) primes. The even integer $6$ can not be expressed as a sum of distinct primes as its only prime representations are $2+2+2$ and $3+3$, thus the condition "sufficiently large" is necessary. If the condition "distinct" is removed, the statement becomes obvious ($n$ copies of $2$).

The contraint to have a representation as a sum of _distinct_ primes does not really affect the magnitude of $r_k(n)$, i.e. the number of ways to write $n$ as a sum of $k$ primes.

Vinogradov's theorem implies that every sufficiently big odd number is the sum of three primes (since it gives a lower-bound for $r_3(n)$), hence every sufficiently big even number is the sum of four distinct primes.