On Lagrange's four squares theorem with almost prime variables
It is conjectured that every sufficiently large integer \(N\equiv 4\pmod{24}\) can be written as the sum of the squares of four primes, but we appear to be far from proving this. It was shown by the reviewer and \textit{D. I. Tolev} [J. Reine Angew. Math. 558, 159--224 (2003; Zbl 1022.11050)] that such \(N\) can be represented as \[ N=x_1^2+x_2^2+x_3^2+p^2 \] with \(p\) prime and each of \(x_1,x_2,x_3\) having at most 101 prime factors. This was improved by \textit{D. I. Tolev} [Bonn. Math. Schr. 360, 17 p. (2003; Zbl 1060.11061)] to 80, and by \textit{Y. Cai} [Int. J. Number Theory 6, No. 8, 1801--1817 (2010; Zbl 1219.11147)] to 42. The first result of the present paper is a further substantial improvement, showing that one can take each of \(x_1,x_2,x_3\) to have at most 5 prime factors. An alternative problem was considered by \textit{J. Brüdern} and \textit{E. Fouvry} [J. Reine Angew. Math. 454, 59--96 (1994; Zbl 0809.11060)], who showed that there are representations \[ N=x_1^2+x_2^2+x_3^2+x_4^2 \] in which each of \(x_1,x_2,x_3,x_4\) has at most 34 prime factors. The papers cited above, by the reviewer and Tolev, by Tolev, and by Cai, improved this successively, replacing 34 by 25, by 20, and by 13 respectively. The present paper makes a further significant improvement, showing that one may take each of \(x_1,x_2,x_3,x_4\) to have at most 4 prime factors. The proof of the first result follows the general structure of the paper by the reviewer and Tolev. However the authors apply Cauchy's inequality at a slightly different point, which turns out to be advantageous. Moreover, rather than use the vector sieve, Chen's reversal of rôles method is employed. For the second result the authors consider \[ N=x_1^2+x_2^2+x_3^2+q^2, \] with \(q\) almost-prime. Rather than use the vector sieve the paper combines a 3-dimensional sieve (for \(x_1,x_2\) and \(x_3\)) with the reversal of rôles method to handle \(q\).
scientific article; zbMATH DE number 6710714
scientific article