Frobenius numbers associated with Diophantine triples of $x^2+y^2=z^r$ (extended version)

Abstract

We give an explicit formula for the $p$-Frobenius number of triples associated with Diophantine equations $x^2+y^2=z^r$, that is, the largest positive integer that can only be represented in $p$ ways by combining the three integers of the solutions of Diophantine equations $x^2+y^2=z^r$. When $r=2$, the Frobenius number has already been given.