Detecting perfect powers by factoring into coprimes

  • Bernstein D
  • Lenstra Jr. H
  • Pila J
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Abstract

This paper presents an algorithm that, given an integer n > 1, finds the largest integer k such that n is a fcth power. A previous algorithm by the first author took time $b^{1 + o(1)} $ where b = lg n; more precisely, time $b\exp (O(\sqrt {\lg b\lg \lg b} ));$ conjecturally, time $b(\lg b)^{O(1)} $ The new algorithm takes time $b(\lg b)^{O(1)} $ . It relies on relatively complicated subroutines--specifically, on the first author's fast algorithm to factor integers into coprimes--but it allows a proof of the $b(\lg b)^{O(1)} $ bound without much background; the previous proof of $b^{1 + o(1)} $ relied on transcendental number theory. The computation of k is the first step, and occasionally the bottleneck, in many number-theoretic algorithms: the Agrawal-Kayal-Saxena primality test, for example, and the number-field sieve for integer factorization.

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APA

Bernstein, D. J., Lenstra Jr., H. W., & Pila, J. (2007). Detecting perfect powers by factoring into coprimes. Mathematics of Computation, 76(257), 385–389. https://doi.org/10.1090/s0025-5718-06-01837-0

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