How big is Graham's number? Too big for the universe

Graham’s number is a finite whole number so large that the observable universe does not contain enough matter to write out its digits, yet mathematicians know it is divisible by 3 and that its last ten digits are 2464195387.

How big is Graham’s number?

Too big for any physical description, so it is built with Knuth’s up-arrow notation, in which each extra arrow repeats the operation before it. 3↑3 is 3 × 3 × 3 = 27. 3↑↑3 is 3↑3↑3, or 3 to the power 27, which is 7,625,597,484,987. 3↑↑↑3 is a tower of powers of 3 that is 7.6 trillion levels high. Then g1 = 3↑↑↑↑3, four arrows, already far beyond anything physical. g2 uses g1 arrows; each step uses the previous number as its arrow count, and g64 is Graham’s number.

That makes it possible to define the number exactly even though it can never be written in full, and any fixed number of its final digits can be computed with modular arithmetic: Graham’s number ends in 7.

Where does Graham’s number come from?

From Ramsey theory. Ronald Graham and Bruce Rothschild asked, in 1971, how many dimensions a cube needs before every two-colouring of the lines joining its corners must contain four corners lying in one plane, all joined in a single colour. A common misconception is that g64 is the bound in their paper. The paper’s bound was a different, smaller number; the g64 version is a slightly larger, easier-to-define number that Graham explained to Martin Gardner, whose Scientific American column made it famous in 1977.

It is also no longer the best answer to its own problem. In 2013 Mikhail Lavrov, Mitchell Lee and John Mackey proved the true answer is below 2↑↑↑6, which is tiny next to even g1. From below, the bound has climbed from 6 in 1971 to 11 in 2003 and 13 in 2008. The true answer lies somewhere between 13 and that new ceiling, and nobody yet knows where.