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Squares of primes and powers of 2. (English) Zbl 0940.11047
Using the circle and sieve methods the authors show that if $$N \equiv 4\bmod 8$$ and $$N$$ is sufficiently large then $$N$$ is representable as a sum of four squares of primes and $$k$$ powers of 2, provided $$k \geq 4$$. They give an asymptotic formula for the number of representations. The analogous result for a sum of two primes and at least three powers of 2 was obtained by Yu. V. Linnik [Tr. Mat. Inst. Steklov 38, 151-169 (1951; Zbl 0049.31402) and Mat. Sb., Nov. Ser. 32, 3-60 (1953; Zbl 0051.03402)].
They also show that a positive proportion of the even numbers not exceeding $$N$$ can be expressed as a sum of two squares of primes and two powers of 2. Here the corresponding result for a sum of a prime and a power of 2 is due to N. P. Romanoff [Math. Ann. 109, 668-678 (1934; Zbl 0009.00801)].

##### MSC:
 11P32 Goldbach-type theorems; other additive questions involving primes 11P05 Waring’s problem and variants 11N36 Applications of sieve methods 11P55 Applications of the Hardy-Littlewood method
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