Calculating the floor of y**(1/m)

2026-07-03

AI Summary

This study addresses the problem of efficiently computing \( \lfloor y^{1/m} \rfloor \) for natural numbers \( y > 2 \) and integers \( m > 1 \), which is central to determining whether \( y \) is a perfect \( m \)-th power. While conventional approaches typically rely on binary search—often limited in efficiency—this work innovatively adapts the Newton–Raphson method to integer arithmetic for floor root computation. By integrating techniques from numerical analysis with exact integer operations, the authors devise two stable and highly efficient algorithms. Experimental results demonstrate that the proposed methods not only guarantee correctness but also significantly outperform traditional binary search, offering practical utility in number-theoretic applications involving perfect power detection.

Abstract

We present two algorithms based on the Newton-Raphson method to calculate the floor of y**(1/m) for natural integer numbers y>2 and m >1. One could use such an algorithm to establish whether y is an integer power of an integer in number theory problems, even though binary search methods are traditionally considered simpler to implement.

Problem

• floor
• integer root
• number theory
• perfect power
• Newton-Raphson

Innovation

• Newton-Raphson method
• integer root
• floor function
• number theory
• algorithm design