Problems related to Waring-Goldbach problem involving cubes of primes

In this note, we try to understand the recent development on the Waring-Goldbach problem involving cubes of primes. Especially, we want to determine whether integers that are either primes, squares of primes, cubes of primes, or a cube of an even number can be written as the sum of four cubes of primes. Meanwhile, we raise some problems that may deepen our understanding of the problem about the sum of four cubes of primes. Moreover, some examples suggest that almost all the cubes of integers can be written as the sum of cubes of four integers.

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