A Newton-Free, Digit-Recurrence Approach to nn-th Root Extraction Using Balanced Numeral Systems

A Newton-Free, Digit-Recurrence Approach ton-th Root Extraction Using Balanced NumeralSystemsBalaji G*November 15, 2025AbstractIntroduction: This paper presents a complete departure from iterativerefinement techniques such as Newton–Raphson for high-precisioncomputation of n-th roots. By leveraging the structural properties of balancednumeral systems, the approach eliminates traditional reliance onfloating-point arithmetic and iterative corrections.Methodology: The framework is divided into two parts:1. Foundational Theory: Lattice completion values and affine-invariantdigit extraction in balanced bases.2. Dual Computation Methods:ˆ Balanced Error Decoding: Direct extraction of digits fromthe balanced representation of the residual N − xp.ˆ Progressive Digit-Appending Method: Structured digitappending via rational approximations, yielding roots withouthigh-precision arithmetic.Execution: Both methods operate in O(log P) steps for P-digit accuracy,support arbitrary odd radices, and emit digits sequentially withoutfull-precision storage. The schemes are carry-free, deterministic, and utilizeonly integer operations.Applications: Enables real-time root extraction for cryptographicprimes, mathematical constants, and embedded systems under constraintsof precision, memory, or side-channel resistance.Conclusion and Significance: This work establishes the first completeNewton-free framework for n-th root computation, combining theoreticalrigor with algorithmic efficiency. It is presented as a practicalalternative to series-based and iterative methods in computationally criticaldomains.