>"Over the past decade, mathematicians have found successively faster multiplication algorithms, each of which has inched closer to n × log n, without quite reaching it.

Then last month, Harvey and van der Hoeven got there.

Their method is a refinement of the major work that came before them. It splits up digits, uses an improved version of the fast Fourier transform, and takes advantage of other advances made over the past forty years. “We use [the fast Fourier transform] in a much more violent way, use it several times instead of a single time, and replace even more multiplications with additions and subtractions,” van der Hoeven said.

Harvey and van der Hoeven’s algorithm proves that multiplication can be done in n × log n steps.

However, it doesn’t prove that there’s no faster way to do it."