Frobenius Numbers Associated with Diophantine Triples of <i>x</i>²-<i>y</i>²=<i>z</i>^r

Abstract

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x2−y2=zr (r≥2), 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 x2−y2=zr. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

Keywords

• Frobenius problem
• Diophantine equations
• Pythagorean triples
• Apéry set