On the Diophantine Equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h

TL;DR

This paper investigates the solutions of a specific exponential Diophantine equation involving powers of 2 and 3, breaking it into manageable cases and solving restricted and generalized versions to understand its solution structure.

Contribution

It introduces a systematic approach to analyze and solve a complex exponential Diophantine equation by breaking it into subcases and solving restricted and generalized forms.

Findings

Identified solutions for restricted subcases.

Solved a generalized form of the equation.

Provided a framework for analyzing similar exponential equations.

Abstract

This paper is a continuation of [1], in which I studied Harvey Friedman's problem of whether the function f(x,y) = x^2 + y^3 satisfies any identities; however, no knowledge of [1] is necessary to understand this paper. We will break the exponential Diophantine equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h into subcases that are easier to analyze. Then we will solve an equation obtained by imposing a restriction on one of these subcases, after which we will solve a generalization of this equation.