Polymorphic Numbers with Exponential Prefix

TL;DR

This paper investigates a special digital property of exponential numbers, generalizing previous work to classify solutions to related equations using advanced mathematical tools.

Contribution

It provides a comprehensive classification and parametrization of solutions to a generalized exponential digital property problem.

Findings

Classified and parametrized solutions to the exponential digital property problem.

Applied advanced mathematical tools like Pell's equation and Fermat's little theorem.

Extended previous partial results to a complete solution set.

Abstract

We observe that the computation $5^2 = 25$ has the digital property of the result being equal to the exponent concatenated directly to the left of the base. The generalization to a Diophantine equation and inequality in number bases has been articulated previously, but a comprehensive answer was not available in the literature. We classify and largely parametrize the solutions. Tools that play key roles are the Newton-Raphson method, the arithmetic-geometric means inequality, Pell's equation, and Fermat's little theorem.