On Waring's Problem IX

Vinogradow has proved not only the amazing result that every large number is the sum of a finite number of nth powers of primes, but has also found out a formula for its upper bound. Let g₁(n) denote the least value of s such that every number from the beginning is the sum of s nth powers of primes or unity. Then it is likely that when n > n₀

g₁(n)=2ⁿ+l-2,

where l=[(3/2)ⁿ].

The object of this paper is to prepare the way for it.