The Fastest Way to Multiply

Everyone learns one way to multiply. For sixty-six years mathematicians have been finding faster ones. This page shows the first shortcut, the records since, and how small the newest claimed step is.

1. The first shortcut

The school method multiplies every digit of one number by every digit of the other. For two 2-digit numbers that is four small multiplications. In 1960 Anatoly Karatsuba, a 23-year-old student, found a way to do it with three. Try any two numbers:

School method

Karatsuba’s method

The trick is in step 3: one multiplication of the two digit-sums, minus the two products already found, gives the middle part that the school method needs two multiplications for. Saving one in four sounds minor. Repeated on the halves of bigger numbers, it compounds.

2. Why one saved multiplication matters

Split a big number in half, and each half in half again, and the saving applies at every level. Drag the slider to make the numbers longer.

Digits in each number:

School

Karatsuba

Counts of single-digit multiplications: digits squared for the school method, and three for every doubling of length for Karatsuba’s. Other bookkeeping is ignored.

3. The ladder of records

Each bar shows roughly how the work grows for two numbers of a million digits. Shorter is faster. The scale is compressed: each step to the left is ten times less work.

These show growth only and ignore constant factors, which in practice decide everything. Real software uses the 1960s and 1971 methods; the 2019 method only wins for numbers far too large to exist, and its authors called it “the end of the road for this problem.” The last two bars are the same length at any scale you could draw.

4. How small is the newest step?

The 2026 paper, written by an OpenAI model, claims a gain measured by a number called κ (kappa). It started at 2⁻¹⁸². Within days a public project had pushed it much higher — each step conditional on the original paper being right. Pick a point on the staircase:

Step:

Figures are from the community project’s own record, ordered from smallest to largest; they are not individually dated. The gain shown is for a number with 10⁸⁰ digits — one for every atom in the observable universe — computed from the stated formula and ignoring constants. The original paper has no machine-checked proof and no specialist has verified it.

A companion visual to UC-329 · Faster Than Anyone Could Notice, a StratIQX case study. It illustrates the subject; it is not part of the case’s scored analysis.

Sources: Karatsuba & Ofman (1962); Schönhage & Strassen (1971); Harvey & van der Hoeven, Annals of Mathematics 193 (2021); OpenAI, “Integer multiplication below n log n” (2026, unverified); community figures from CrocSwap/integer-mult-bounds, read October 14, 2026.

See also: Colouring the Plane, the companion to UC-328.