In a paper by MELVYN B. NATHANSON (https://arxiv.org/abs/1401.7598), it was claimed that: The Goldbach conjecture implies that the set of primes is an asymptotic basis of order 3.
My question is: How this is possuble
Also, what is the order of the set of even numbers if the Goldbach conjecture is true.
The primes being an asymptotic basis of order 3 means that every sufficiently large integer is a sum of exactly three primes. Goldbach's conjecture tells us that every sufficiently large even integer is a sum of two primes. (Here "sufficiently large" means at least 4.)
Thus, assuming Goldbach, every even integer $n \ge 6$ is a sum of exactly three primes (by writing $n-2$ as a sum of two primes and adding 2) and every odd integer $n \ge 7$ is a sum of exactly three primes (by writing $n-3$ as a sum of two primes and adding 3.)