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For $n\ge 1$, let $r_4(n)$ count the integer quadruples $(a,b,c,d)$ with $a^2+b^2+c^2+d^2=n$, counting order, signs, and zeros. Jacobi's four-square theorem givesThe right side is positive, so this also implies that every positive integer is a sum of four squares.

For $n=1$, the formula gives eight representations: choose the position of $\pm 1$. For $n=2$, it gives $8(1+2)=24$, matching the six choices of two nonzero positions and four sign choices. Specifying the counting convention matters: unordered nonnegative representations have a different count.