Polymorphic Numbers with Exponential Prefix

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.