Frobenius Numbers Associated with Primitive Pythagorean Quadruples

Let $(a,b,c,d)=(2mn,2mp,m^2-n^2-p^2,m^2+n^2+p^2)$ be a primitive Pythagorean quadruple and let $S=⟨a,b,c,d⟩$ be the numerical semigroup generated by $a,b,c,$ and $d\mathrm{.}$ For convenience, we also let $Q=n^2+p^2,\delta =\gcd (n,p),$ and $n=\delta n_0$ for some $n_0\in \mathbb{Z}$. In this paper, we determine the Frobenius number of $S$ and derive an explicit formula in terms of $m$, $n$, and $p$ with the assumption that $m\ge 2Q$ for $\delta =1$ and $m\ge \frac{2Q}{\delta }-1$ for the remaining cases. The proof is based on an explicit complete residue system modulo $2mn$, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.