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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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