The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Factoring into coprimes in essentially linear time
Journal of Algorithms
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Let N be a positive integer, and let &sgr;(N) denote the sum of the positive integral divisors of N. We show computing &sgr;(N) is equivalent to factoring N in the following sense: there is a random polynomial time algorithm that, given &sgr;(N), produces the prime factorization of N, and &sgr;(N) can be easily computed given the factorization of N. We show that the same sort of result holds for &sgr;k(N), the sum of the k-th powers of divisors of N. We give three new examples of problems that are in Gill's complexity class BPP: {perfect numbers}, {multiply perfect numbers}, and {amicable pairs}. These are the first “natural” candidates for BPP - RP.