 Jacobi's four-square theorem

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.

Jacobi's four-square theorem is an extension of Lagrange's four-square theorem, which states that every positive integer can be expressed as the sum of four squares. Jacobi's contribution to this area lies in his work on representing numbers as sums of squares and his formulation of a more explicit representation. The theorem states that the number of ways to represent a natural number \( n \) as a sum of four squares can be expressed through a specific counting function.