Algorithmica

, Volume 9, Issue 4, pp 313–328

# Sieve algorithms for perfect power testing

• Eric Bach
• Jonathan Sorenson
Article

## Abstract

A positive integern is a perfect power if there exist integersx andk, both at least 2, such thatn=xk. The usual algorithm to recognize perfect powers computes approximatekth roots fork≤log2n, and runs in time O(log3n log log logn).

First we improve this worst-case running time toO(log3n) by using a modified Newton's method to compute approximatekth roots. Parallelizing this gives anNC2 algorithm.

Second, we present a sieve algorithm that avoidskth-root computations by seeing if the inputn is a perfectkth power modulo small primes. Ifn is chosen uniformly from a large enough interval, the average running time isO(log2n).

Third, we incorporate trial division to give a sieve algorithm with an average running time ofO(log2 n/log2 logn) and a median running time ofO(logn).

The two sieve algorithms use a precomputed table of small primes. We give a heuristic argument and computational evidence that the largest prime needed in this table is (logn)1+O(1); assuming the Extended Riemann Hypothesis, primes up to (logn)2+O(1) suffice. The table can be computed in time roughly proportional to the largest prime it contains.

We also present computational results indicating that our sieve algorithms perform extremely well in practice.

### Key words

Perfect powers Number theoretic algorithms Riemann hypothesis Newton's method Sieve algorithms Parallel algorithms Average-case analysis

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