Goldbach-Linnik type problems involving unequal powers of primes and powers of 2

Abstract

It is proved that every pair of sufficiently large even integers can be represented as not only a pair of one prime, one square of prime, one cube of prime, one biquadrate of prime and $34$ powers of $2$, but also a pair of two squares of primes, two cubes of primes, two biquadrates of primes and $36$ powers of $2$.