Solvability of the Diophantine Equation 𝑥2+𝑝2 = 𝑦n when 𝑝 is a Prime

Abstract

This paper deals with solving a Diophantine equation in the type of 𝑥² + 𝑦² = z by using mainly the unique factorization in the ring of Gaussian integers and some elementary facts in number theory. We have known that such an equation has no any integer solution apart from the trivial one when 𝑛 is an even positive integer greater than 2, and it has known as the “Pythagorean equation” in the particular case 𝑛 = 2, which can however be completely solved. Our interest here is to naturally focus on the odd positive one and the unknown variable 𝑦 (or 𝑥) is especially restricted to be a prime number 𝑝 only. Eventually, it turns out that we are able to give necessary and sufficient conditions for having an integer solution of such an equation in the case 𝑛 = 3.