A Novel Summation Formula for the Hurwitz-Kronecker Class Number

The purpose of this paper is to present a novel and elegant summation formula for $H_w$, the Kronecker-Hurwitz class number. Specifically, for any prime $p$, we have the formula: $$\sum _{t^2<p}{H}_w(t^2-p)=\frac{p-2}{3}\mathrm{.}$$