A Fast Algorithm for Determining the Existence and Value of Integer Roots of N

TL;DR

This paper introduces an efficient integer arithmetic-based algorithm to determine if a number is a perfect square and find its root, with potential extensions to arbitrary roots and prime product identification.

Contribution

The paper presents a novel four-step deterministic algorithm for identifying perfect squares and their roots without multiplication or division by large integers.

Findings

Algorithm scales approximately as log(sqrt(N)/2)

Certifies non-squares when no solution is found

Extends methods to arbitrary root finding and prime product detection

Abstract

We show that all perfect odd integer squares not divisible by 3, can be usefully written as sqrt(N) = a + 18p, where the constant a is determined by the basic properties of N. The equation can be solved deterministically by an efficient four step algorithm that is solely based on integer arithmetic. There is no required multiplication or division by multiple digit integers, nor does the algorithm need a seed value. It finds the integer p when N is a perfect square, and certifies N as a non-square when the algorithm terminates without a solution. The number of iterations scales approximately as log(sqrt(N)/2) for square roots. The paper also outlines how one of the methods discussed for squares can be extended to finding an arbitrary root of N. Finally, we present a rule that distinguishes products of twin primes from squares.