Frobenius numbers of Pythagorean triples

Given relatively prime integers $a_1,\dots ,a_n$, the Frobenius number $g(a_1,\dots ,a_n)$ is defined as the largest integer which cannot be expressed as $x_1{a}_1+\cdots +x_n{a}_n$ with $x_i$ nonnegative integers. In this article, we give the Frobenius number of primitive Pythagorean triples. That is, $$g(m^2-n^2,2mn,m^2+n^2)=(m-1)(m^2-n^2)+(m-1)(2mn)-(m^2+n^2)\mathrm{.}$$