Refining Lagrange's four-square theorem

Lagrange's four-square theorem asserts that any n is an element of N = {0,1,2,...} can be written as the sum of four squares. This can be further refined in various ways. We show that any n is an element of N can be written as x(2)+y(2)+z(2) +w(2) with x,y,z,w is an element of Z such that x+y+z (or x+2y, or x+y+2z) is a square (or a cube). We also prove that any n is an element of N can be written as x(2)+y(2)+z(2)+w(2) with x,y,z,w is an element of N such that P(x,y,z) is a square, whenever P(x,y,z) is among the polynomials x, 2x, x-y, 2x-2y, a(x-y(2)) (a = 1, 2, 3), x(2)-3y(2), 3x(2)-2y(2), x(2)+ky(2) (k= 2,3,5,6,8,12), (x+4y+4z)(2)+(9x+3y+3z)(2), x(2)y(2)+y(2)z(2)+z(2)x(2), x(4)+8y(3)z+8yz(3), x(4)+16y(3)z+64yz(3). We also pose some conjectures for further research; for example, our 1-3-5-Conjecture states that any n E N can be written as x(2)+y(2)+z(2)+w(2) with x,y,z,w is an element of N such that x+3y+5z is a square. (C) 2017 Elsevier Inc. All rights reserved.