tl;dr: The grade-school multiplication algorithm runs in O(n²), long believed to be optimal until 23-year-old Anatoly Karatsuba disproved this in 1960 with an O(n^1.585) algorithm that trades multiplications for cheaper additions—still used today in languages like Python for large integers. In 2019, Harvey and van der Hoeven achieved O(n log n), suspected to be the theoretical speed limit, though their algorithm is "galactic" (only faster for impractically huge numbers). Proving O(n log n) is truly optimal remains an open problem.