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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.02703 |
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| _version_ | 1866917186452324352 |
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| author | Pareth, Suresan |
| author_facet | Pareth, Suresan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02703 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact Constructive Digit-by-Digit Algorithms for Integer $e$-th Root Extraction Pareth, Suresan Symbolic Computation 68W40, 11Y16, 65D20, 03F60 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. |
| title | Exact Constructive Digit-by-Digit Algorithms for Integer $e$-th Root Extraction |
| topic | Symbolic Computation 68W40, 11Y16, 65D20, 03F60 |
| url | https://arxiv.org/abs/2601.02703 |