LAGRANGE'S FOUR SQUARES THEOREM WITH VARIABLES OF SPECIAL TYPE

Let N denote a sufficiently large integer satisfying N ≡ 4 (mod 24), and Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that the equation [Formula: see text] is solvable in one prime and three P42, or in four P13. These results constitute improvements upon that of Heath-Brown and Tolev.