Squares of primes and powers of 2
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 \textit{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 \textit{N. P. Romanoff} [Math. Ann. 109, 668-678 (1934; Zbl 0009.00801)].
sum of four squares of primes and powers of two
sum of two squares of primes and two powers of two
scientific article; zbMATH DE number 1395806