Diophantine Analysis of a Digital Anomaly

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 $k=2$.