Exact Constructive Digit-by-Digit Algorithms for Integer $e$-th Root Extraction

Abstract

We present a unified constructive digit-by-digit framework for exact root extraction using only integer arithmetic. The core contribution is a complete correctness theory for the fractional square root algorithm, proving that each computed decimal digit is exact and final, together with a sharp truncation error bound of $10^{-k}$ after $k$ digits. We further develop an invariant-based framework for computing the integer $e$-th root $\lfloor N^{1/e} \rfloor$ of a non-negative integer $N$ for arbitrary fixed exponents $e \ge 2$, derived directly from the binomial theorem. This method generalizes the classical long-division square root algorithm, preserves a constructive remainder invariant throughout the computation, and provides an exact decision procedure for perfect $e$-th power detection. We also explain why exact digit-by-digit fractional extraction with non-revisable digits is structurally possible only for square roots ($e=2$), whereas higher-order roots ($e \ge 3$) exhibit nonlinear coupling that prevents digit stability under scaling. All proofs are carried out in a constructive, algorithmic manner consistent with Bishop-style constructive mathematics, yielding explicit algorithmic witnesses, decidable predicates, and guaranteed termination. The resulting algorithms require no division or floating-point operations and are well suited to symbolic computation, verified exact arithmetic, educational exposition, and digital hardware implementation.