Frobenius numbers of Pythagorean triples

Abstract

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, \[ g(m^2-n^2, 2mn, m^2+n^2) = (m-1)(m^2-n^2) + (m-1)(2mn) - (m^2 + n^2). \]

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