Restricted sums of four squares

摘要

We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including 1). For example, we show that each n = 1, 2, 3, ... can be written as x(2)+y(2)+z(2)+w(2) (x, y, z, w is an element of N = {0, 1, 2, ...}) with vertical bar x+y-z vertical bar is an element of{4(k): k is an element of N} (or vertical bar 2x - y vertical bar is an element of{4(k):k is an element of N},or x + y - z is an element of {+/- 8(k): k is an element of N}boolean OR{0}subset of{t(3) :t is an element of Z}), and that we can write any positive integer as x(2) + y(2) + z(2) + w(2) (x,y,z,w is an element of Z) with x + y + 2z (or x + 2y + 2z) a power of four. We also prove that any n is an element of N can be written as x(2) + y(2) + z(2) + 2w(2) (x, y, z, w is an element of Z) with x+ y + z + w a square (or a cube). In addition, we pose some open conjectures for further research; for example, we conjecture that any integer n > 1 can be written as a(2) + b(2) + 3(c) + 5(d) with a, b, c, d is an element of N.