Does there exist an integer $s$ such that every integer $> 1$ can be written as a sum of at most $s$ primes?

Does there exist an integer $s$ such that every integer $> 1$ can be written as a sum of at most $s$ primes ?

Terrence Tao has proven that every odd number greater than one is the sum of at most five primes. Hence, using $3$ as a prime that we can substract to any even number greater than $5$, and noticing that $2$ is prime and $4=2+2$, every odd number is the sum of at most six primes.
Of course, this is a very loose approximation. If it is proven, Goldbach's conjecture would allow us to lower the number of needed primes to $3$.