Waring's Problem for Powers of Primes II

In a recent paper, Pillai has examined one of my conjectures and proved the

THEOREM. If l ≤r ≤ 2ⁿ-2l-2 and n ≥ 20, then every integer not exceeding N is the sum of at most I nth powers of primes ≥ 1,

where

N = c^(1.6)n/6.31ⁿ,

3ⁿ = 1.2^r+r,

and

I = 2ⁿ+l-2.

He has not stated how far his theorem holds for values of n < 20. The object of the present paper is to study the validity of Pillai's theorem when n ≤ 19.