On Lagrange's four squares theorem with almost prime variables

Abstract

In 1994, Brüdern and Fouvry [1] initiated the investigation of Lagrange’s four squares theorem with almost prime variables. In this paper, we prove that every sufficiently large integer, congruent to 4 modulo 24, can be represented as a sum of four squares of integers, each of which has at most four prime factors. Instead of the four-dimensional vector sieve developed by Brüdern and Fouvry [1], we establish this result by combining the three-dimensional sieve and the switching principle.