Diophantine Analysis of a Digital Anomaly

TL;DR

This paper investigates a digital anomaly related to division in various number bases by analyzing an exponential Diophantine equation, providing parametrizations, finiteness results, and conjectures on solutions.

Contribution

It generalizes the digital anomaly to a broad Diophantine framework, offering parametrizations and finiteness proofs using advanced number theory tools.

Findings

All solutions are finitely many per base due to Baker's theorem.

Infinite families of solutions exist when numerator and denominator are not coprime.

Conditional bounds on solutions with fixed digit length are proposed.

Abstract

The arithmetic-digital anomaly of $5\div 2 = 2.5$ has been observed several times in the past. We generalize it to an exponential Diophantine equation and inequality in the general number base, which is the object of our analysis. First, we produce a near-parametrization of all solutions using a modification of the standard parametrization of Pythagorean triples. We use this parametrized function to find all solutions where the numerator and denominator are coprime, and we construct infinite families where they are not coprime. Next, we use a variant of Baker's theorem from transcendental number theory to prove that each number base admits only finitely many solutions. Lastly, we use the $abc$ conjecture to conditionally show that only finitely many solutions have a numerator with $k$ digits, for each $k\ge 3$. A conjecture is offered for $k=2$.