Abstract
It is proved that every pair of sufficiently large even integers can be represented in the form of a pair of equations of eight prime cubes and 609 powers of 2, and each sufficiently large even integer is the sum of eight cubes of primes and 157 powers of 2. These results constitute refinements upon those of Z. X. Liu (2013) and of X. D. Zhao and W. X. Ge (2020).
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