Representation of even integers as sums of squares of primes and powers of 2
It has been shown by \textit{J. Liu}, \textit{M.-C. Liu} and \textit{T. Zhan} [Monatsh. Math. 128, 283-313 (1999; Zbl 0940.11047)] that there is an integer \(k\) such that every sufficiently large even number may be written in the form \[ p^2_1+p^2_2+ p^2_3+p^2_4+ 2^{e_1}+ \cdots+ 2^{e_k}. \] Here it is shown that one may be take \(k=8330\). There are a couple of significant departures from the work of Liu, Liu and Zhan. Most notably, one has to estimate the number of solutions of the equation \[ n=p^2_1+ p^2_2- p^2_3- p^2_4 \] for a given non-zero integer \(n\), and the present paper uses the `vector sieve' of \textit{J. Brüdern} and \textit{E. Fouvry} [J. Reine Angew. Math. 454, 59-96 (1994; Zbl 0809.11060)]. Although this is a clear improvement, a more direct approach is clearly desirable.
sums of squares of primes and powers of 2
scientific article; zbMATH DE number 1515455
scientific article