In this article we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of two or three squares of primes and a kth power of prime for any integer k ≥ 2. For example, we prove that with at most0(N~1-(1/3k2~(k-2))+exceptions for k ≥ 4, all positive inte- gers n ≤ N, satisfying the necessary congruence conditions, are the sum of two squares of primes and a kth of prime. This improves substantially the previous results in this direction.
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