A deterministic algorithm for finding $r$-power divisors

TL;DR

This paper introduces a deterministic algorithm that efficiently finds all r-power divisors of an integer N, improving the time complexity for problems like testing squarefreeness compared to previous methods.

Contribution

The paper presents a new deterministic algorithm with provable time bounds for finding r-power divisors, extending prior work and improving efficiency for specific number-theoretic problems.

Findings

Algorithm finds all r-power divisors in time O(N^{1/4r+ε})

Can test squarefreeness of N in time O(N^{1/8+ε})

Improves upon previous bounds for related problems

Abstract

Building on work of Boneh, Durfee and Howgrave-Graham, we present a deterministic algorithm that provably finds all integers $p$ such that $p^r \mathrel| N$ in time $O(N^{1/4r+\epsilon})$ for any $\epsilon > 0$. For example, the algorithm can be used to test squarefreeness of $N$ in time $O(N^{1/8+\epsilon})$; previously, the best rigorous bound for this problem was $O(N^{1/6+\epsilon})$, achieved via the Pollard--Strassen method.