Squares of primes and powers of 2.

*(English)*Zbl 0940.11047Using 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)].

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)].

Reviewer: George Greaves (Cardiff)

##### 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 |