PAPPADIA THEOREM - A UNIVERSAL METHOD FOR CALCULATING DECIMAL PERIODS OF FRACTIONS

We present the Pappadia Theorem for the direct calculation of the period and antiperiod of rational fractions in their decimal representation. The theorem establishes that for a fraction p/q in reduced form with q = 2^a · 5^b · M where gcd(M, 10) = 1, the antiperiod length is μ = max(a, b) and the period length is λ = ord_M(10), where ord_M(10) denotes the multiplicative order of 10 modulo M. The algorithmic implementation achieves computational complexity O(log M) versus the traditional complexity O(λ), obtaining performance improvements of 10-100× for fractions with long periods. The repository contains the reference Python implementation of the theorem, with complete validation across different categories of fractions. While this document presents the fundamental algorithm, commercial implementations with additional optimizations (Pappadia Booster) are available under commercial license. © 2024-2025 Umberto Junior Pappadia - All rights reservedSIAE Deposit No. 2025/00839 (Deposited: 16/04/2025)PAPPADIA® is a registered trademark at EUIPO