Waring's Theorem for Powers of Primes

The ideal Waring's Theorem stales that every positive integer is a sum of I integral nth powers ≥ 0, where

I = [(3/2)ⁿ]+2ⁿ-2.

Chowla and the author have considered the corresponding problem for primes so far as the squares and cubes are concerned. It has been shown that all positive integers i ≤ 240,000 are sums of at the most eight squares of primes p ≥ 1. It has also been shown that all positive integers i ≤ 100,000; i ≠ 1301 can be expressed as the sums of at the most twelve cubes of primes; while 1301 requires 13 and no less. In fact it is noticed that integers greater than 8161 require 11 cubes of primes at the most.