Sieve algorithms for perfect power testing
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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.
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- Sieve algorithms for perfect power testing
Volume 9, Issue 4 , pp 313-328
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- Perfect powers
- Number theoretic algorithms
- Riemann hypothesis
- Newton's method
- Sieve algorithms
- Parallel algorithms
- Average-case analysis
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