Representations as sums of four squares

Let $n$ be a nonnegative integer. A representation of $n$ as a sum of four squares is an ordered quadruple $(a,b,c,d)\in {\mathbb{Z}}^4$ with $n=a^2+b^2+c^2+d^2$. The integer $n$ is a sum of four integer squares when at least one such quadruple exists.

Two representations $(a,b,c,d)$ and $(a^{\mathrm{\prime }},b^{\mathrm{\prime }},c^{\mathrm{\prime }},d^{\mathrm{\prime }})$ of the same $n$ are equivalent up to signs and order when one is obtained from the other by permuting the four coordinates and independently changing the sign of any of them, and essentially different when they are not so equivalent. Since changing a sign does not change an absolute value and permuting the coordinates permutes them, two representations are equivalent exactly when the multisets $\{\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid },\mathrm{\mid }c\mathrm{\mid },\mathrm{\mid }d\mathrm{\mid }\}$ and $\{\mathrm{\mid }{a}^{\mathrm{\prime }}\mathrm{\mid },\mathrm{\mid }{b}^{\mathrm{\prime }}\mathrm{\mid },\mathrm{\mid }{c}^{\mathrm{\prime }}\mathrm{\mid },\mathrm{\mid }{d}^{\mathrm{\prime }}\mathrm{\mid }\}$ coincide.

The coordinates range over all of $\mathbb{Z}$. They may be zero, they may be negative, and they may repeat. So an integer written as a sum of one, two or three squares is a sum of four squares as well, its remaining coordinates being $0$; and $-1$ is an admissible coordinate wherever $1$ is, with the same square. This matters where the identity of Euler's four-square product identity is applied, since its output coordinates are frequently negative or zero even when its inputs are not.

Only nonnegative $n$ is defined. A square is nonnegative and a sum of four nonnegative integers is nonnegative, so no negative integer admits a quadruple and the definition would be vacuous there.

The value $0$. A sum of four squares vanishes exactly when each square vanishes, so $0=0^2+0^2+0^2+0^2$ and $(0,0,0,0)$ is the only representation of $0$. Every representation of a positive integer therefore has at least one nonzero coordinate.

Agreement with the two-square convention. The convention here is the one Representations and primitive representations as sums of two squares uses for pairs: a representation is an ordered tuple, so order and sign are recorded, and equivalence up to signs and order is what quotients them out. The only change is the number of coordinates. That every nonnegative integer has at least one representation is Lagrange's four-square theorem: every nonnegative integer is a sum of four integer squares; the definition itself asserts nothing about existence.