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Squares of primes and powers of 2. II. (English) Zbl 0997.11082
In [Monatsh. Math. 128, 283-313 (1999; Zbl 0940.11047)] the present authors established a positive lower bound of the presumably correct order of magnitude for a count of the number of representations of a large number \(N \equiv 4 \bmod 8\) as a sum of four squares of primes and \(k\) powers of 2. The principal theorem of this paper holds nominally when \(k\geq 4\), but the presence of \(O\)-terms depending on \(k\) mean that the result is non-trivial only when \(k\) is sufficiently large. Subsequently, the first two authors showed in [J. Number Theory 83, 202-225, (2000; Zbl 0961.11035)] that \(k=8330\) was admissible. The counting operation was performed using suitable weights expressed in terms of the von Mangoldt function.
In the present paper a corresponding asymptotic formula is obtained when \(N \equiv 4 \bmod 24\) and \(k\) is sufficiently large. This is deduced from a corresponding mean square result, which also implies that almost all \(n \equiv 2 \bmod 24\) can be expressed as a sum of two squares of primes and a bounded number of powers of 2.
The method now used draws on the dispersion method in a manner used by P. X. Gallagher [Invent. Math. 29, 125-142 (1975; Zbl 0305.10044)] on the analogous question about sums of primes and powers of 2.

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
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