Sums of three squares under congruence condition modulo a prime

Let p be an odd prime. We show that the integral points on the sphere with radius n are equidistributed modulo p   as n⟶∞n⟶∞ where n   is not of the shape 4l(8m+7)4l(8m+7) and its 2-adic valuation is bounded. In particular if n is sufficiently large and if n   satisfies a congruence equation α12+α22+α32≡n(modp) where p2|np2|n if all αi≡0(modp), then there are integers xixi with xi≡αi(modp)(i=1,2,3) satisfying x12+x22+x32=n. The similar result holds also in the case modulo 8.