We refine Lagrange’s four-square theorem in new ways by imposing some restrictions involving powers of two (including [Formula: see text]). For example, we show that each [Formula: see text] can be written as [Formula: see text] ([Formula: see text] with [Formula: see text] (or [Formula: see text], or [Formula: see text]), and that we can write any positive integer as [Formula: see text] with [Formula: see text] (or [Formula: see text]) a power of four. We also prove that any [Formula: see text] can be written as [Formula: see text] [Formula: see text] with [Formula: see text] a square (or a cube). In addition, we pose some open conjectures for further research; for example, we conjecture that any integer [Formula: see text] can be written as [Formula: see text] with [Formula: see text].