Proof of a conjecture of Cooper

Abstract: In this note, we confirm a conjectural formula on the number of representations of the square of an odd prime by a sum of an odd number of squares, i.e., \(\#\{(x_{1},\ldots,x_{2k+1})\in\mathbb{Z}^{2k+1}:\,x_{1}^{2}+\cdots+x_{2k+1}^{2}=p^{2}\},\) proposed by Cooper.