Integer solutions of $a^2+ab+b^2=7^n$

TL;DR

This paper presents two distinct proofs demonstrating the existence of coprime positive integers satisfying the equation a^2+ab+b^2=7^n for any positive integer n.

Contribution

It provides novel proofs establishing the existence of solutions to a specific exponential Diophantine equation.

Findings

Existence of coprime solutions for all n

Two different proof methods provided

Clarifies the structure of solutions to the equation

Abstract

In this article we will show $2$ different proofs for the fact that there exist relatively prime positive integers $a,b$ such that: $a^2+ab+b^2=7^n$.