Perfect Powers of Five with Few Ternary Digits

TL;DR

This paper analyzes a specific Diophantine equation related to powers of five and ternary digits, proving insolubility in most cases and identifying a unique solution under certain conditions.

Contribution

It establishes new results on the insolubility of the equation for various parity cases of the exponents, and confirms the uniqueness of a known solution.

Findings

The equation is insoluble when both exponents are even.

The equation has a unique solution when both exponents are odd.

No solutions exist for large values of n within the specified bounds.

Abstract

In this note we will analyze a diophantine equation raised by Michael Bennett in [1] that is pivotal in establishing that powers of five has few digits in its ternary expansion. We will show that the Diophantine equation $3^{a}+3^{b}+2=n^5$, where $(n,3)=1$ and $a>b>0$ is insoluble for pairs of positive integers $(a,b)$ where they are both even or one is even and the other is odd. In the case where both $(a,b)$ are odd, there is one known solution $2^5=3^3+3^1+2.$ We will show that there are no other solutions to the diophantine equation for $n^{5}<32\left(1+3(10^6)\right)^5$.