Sums of divisors, perfect numbers, and factoring (extended abstract)

Abstract

Let N be a positive integer, and let σ(A) denote the sum of the positive integral divisors of N. We show computing σ(N) is equivalent to factoring N in the following sense: there is a random polynomial time algorithm that, given σ(N), produces the prime factorization of N, and σ(N) can be easily computed given the factorization of N. We show that the same sort of result holds for σ(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.