Prime Factorization

Find a much faster factorization algorithm.

Unsolved Number Theory
Breakthrough

Contributor: Epoch Staff

About the problem

This problem asks for an algorithm that can factor semiprimes \(N = p \cdot q\) faster than the best known classical methods can. The tests leave a large enough margin that any positive score is very likely to be an algorithmic breakthrough.

Warm-up: we ask for a program that can handle numbers of a size that is doable with existing techniques and software.

AI prompts

The difficulty level can be tuned by setting the size of number to factor. 90 digits should be doable just using existing software. 150 digits is likely in "breakthrough" territory.

Warm-up

Produce an algorithm that can factor balanced 90-digit semiprimes on a typical modern laptop in under ten minutes.

Full problem

Produce an algorithm that can factor balanced 150-digit semiprimes on a typical modern laptop in under ten minutes.

Mathematician survey

The author assessed the problem as follows.

Number of mathematicians highly familiar with the problem:

a majority of those working in a subfield (≈100)

Number of mathematicians who have made a serious attempt to solve the problem:

10–50

Rough guess of how long it would take an expert human to solve the problem:

3–10 years

Notability of a solution:

a major advance in a broader field; one of the best results of the year

A solution would be published:

in a leading general journal

Likelihood of a solution generating more interesting math:

very likely: any solution would lead to a rich set of follow-on work

Probability that the problem is solvable as stated:

80–95%