ISSN:
1432-0541
Keywords:
Perfect powers
;
Number theoretic algorithms
;
Riemann hypothesis
;
Newton's method
;
Sieve algorithms
;
Parallel algorithms
;
Average-case analysis
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract A positive integern is a perfect power if there exist integersx andk, both at least 2, such thatn=x k . The usual algorithm to recognize perfect powers computes approximatekth roots fork≤log 2 n, and runs in time O(log3 n log log logn). First we improve this worst-case running time toO(log3 n) by using a modified Newton's method to compute approximatekth roots. Parallelizing this gives anNC 2 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(log2 n). 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01228507
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