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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
##### Keywords:
Goldbach problem; almost Goldbach; dispersion method
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##### References:
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