# Solutions of the Exponential Equation 7x^2 + 59y^2 = 3^m. A simple Algorithm producing all the primitive solutions

## TL;DR

This paper presents an elementary arithmetic-based method and a simple algorithm to find all primitive positive solutions of the exponential Diophantine equation 7x^2 + 59y^2 = 3^m, which can be generalized to similar equations.

## Contribution

It introduces a novel elementary approach and an efficient algorithm for solving a specific exponential Diophantine equation, avoiding radicals or complex numbers.

## Key findings

- Algorithm generates all primitive solutions
- Method applies to a class of similar equations
- No radicals or complex numbers used

## Abstract

We provide a method, using essentially elementary arithmetic, to solve the exponential diophantine equation 7x^2 + 59y^2 = 3^m, which leads to a simple algorithm, with no use of radicals or complex numbers, that generates all the (infinitely many) primitive and positive solutions of this equation. The method can be generalized to solve a class of exponential equations of the form ax^2 + by^2 = ck^z.