# Gauss - Dirichlet class number formula

Let $$p=8k+3$$ be a prime. Then the class number of the imaginary quadratic field $$\mathbb Q(\sqrt{-p})$$ is given by $$h(-p)=\frac 13\sum_{k=1}^{\frac{p-1}{2}}\left(\frac kp \right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $$L$$-series at $$1$$ with the class number, but rather about its particular corollary).

Let $$p=2n+1$$ be a prime with $$p\equiv3\pmod4$$. By Wilson's theorem, $$(n!)^2\equiv1\pmod p$$ and hence $$n!\equiv\pm1\pmod p$$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$\left(2-\left(\frac 2p\right)\right) h(-p)=\sum_{k=1}^n\left(\frac kp\right)$$ to deduce in few lines that $$n!\equiv(-1)^{(h(-p)+1)/2}\pmod p$$ if $$p>3$$. This is a nice application of Dirichlet's class number formula.