2025/12/05 by Samer Seraj, Seraj, Samer · 1 voice
Computer Science · Mathematics · Physics and Astronomy · #11A63 #11D09 #11D61 #11J86 #Advanced Mathematical Theories and Applications #Algebraic Geometry and Number Theory #Digital Image Processing Techniques #FOS: Mathematics #History and Overview (math.HO) #math.HO
paper · pdf · doi:10.48550/arxiv.2512.06056
openalex publication_date 2025/12/05 · arxiv published 2025/12/05 · arxiv updated 2025/12/05 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28
The arithmetic-digital anomaly of 5÷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≥ 3. A conjecture is offered for k=2.