Restricted sums of four squares

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Abstract
We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two. For example, we show that any positive integer can be written as $x^2+y^2+z^2+w^2\ (x,y,z,w𝟄\mathbb Z)$ with $x+y+z+w=2^{⌊(\mbox{ord}_2(n)+1)/2⌋}$, and that each positive integer can be written as $x^2+y^2+z^2+w^2\ (x,y,z,w𝟄\mathbb Z)$ with $x+y+z$ (or $x+y+2z$, or $x+2y+2z$) a power of four (including 1). We also pose some open conjectures; for example, we conjecture that any positive integer can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w$ nonnegative integers such that $x+2y-2z$ is a power of four, and that any positive integer can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w$ integers such that $x+2y+4z+8w𝟄\{8^k:\ k=0,1,2,…\}$. ; Comment: 17 pages