What is the fastest integer factorization algorithm?

Question

I\'ve written a program that attempts to find Amicable Pairs. This requires finding the sums of the proper divisors of numbers.

Here is my current sumOfDiviso

Answer 1

This is an important open mathematical problem as of 2020

Others have answered from a practical point of view, and there is high probability that for the problem sizes encountered in practice, that those algorithms are close to the optimal.

However, I would also like to highlight, that the more general mathematical problem (in asymptotic computation complexity, i.e. as the number of bits tends to infinity) is completely unsolved.

No one has ever been able to prove what is the minimal optimal time for of what is the fastest possible algorithm.

This is shown on the Wikipedia page: https://en.wikipedia.org/wiki/Integer_factorization The algorithm also figures on Wiki's "List of unsolved problems in computer science" page: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science

All that we know is that the best we currently have is the general number sieve. And until 2018, we didn't even have a non-heuristic proof for its complexity.

As of 2020, we haven't even proved if the problem is NP-complete or not (although it is obviously NP since all you have to do to verify a solution is to multiply the numbers)! Although it is widely expected for it to be NP-complete. We can't be that bad at finding algorithms, can we?