Abstract
In this paper, it is proved that for [Formula: see text], every sufficient large odd integer is a sum of one prime, two squares of primes and [Formula: see text] powers of two. Furthermore, for [Formula: see text], every pair of large odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two squares of primes and [Formula: see text] powers of two. These improve the previous results [Formula: see text] and [Formula: see text].
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