Graham’s Number

Graham’s number is a number invented by Ronald Graham. In order to explain what it is, the notation must be understood. It’s called up-arrow notation, denoted by the ↑ symbol. One up-arrow just denotes that the second number is an exponent. For example, 3↑3 is 3^3, or 27. Using two arrow creates the fourth thing in the sequence of addition, multiplication, and exponentiation. Some call this math operation tetration. 3↑↑3 is 3^(3^3), 3^27, or 7,625,597,484,987. Using a third arrow, you can probably predict what happens. 3↑↑↑3 is 3↑↑(3↑↑3), or 3↑↑7,625,597,484,987. This means that you have (3^(3^(3^(...(3^3)...)))), and there are 7,625,597,484,987 3’s. For perspective, 3↑↑4, or 3^7,625,597,484,987, contains 3,638,334,640,024 digits. I’m not kidding, that is the actual number of digits, compute it using the Big Online Calculator. And yet, despite how far blown out of proportion this thing has been, it’s still not large enough. We need a fourth arrow. Don’t even get me started on the size of 3↑↑↑↑3, or 3↑↑↑(3↑↑↑3). And that number is called G(1). G(2) is 3↑↑↑...↑↑↑3. There are G(1) arrows. G(3) is 3↑↑↑...↑↑↑3, with G(2) 3’s. You get it now? Graham’s number is defined as G(64). And despite its immense size, it actually has a purpose. Suppose you had higher-dimensional hypercubes, and you had two colors for edges, and you wanted to know how many dimensions it took before a square where all lines were the same color was forced. The upper bound on that answer is Graham’s number.