Knuth's Up-arrow Notation - Generalizations

Generalizations

Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an n-arrow operator is useful (and also for descriptions with a variable number of arrows), or equivalently, hyper operators.

Some numbers are so large that even that notation is not sufficient. Graham's number is an example. The Conway chained arrow notation can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.

$\begin{matrix} a\uparrow^n b & = & \mbox{hyper}(a,n+2,b) & = & a\to b\to n \\ \mbox{(Knuth)} & & & & \mbox{(Conway)} \end{matrix}$

It is generally suggested that Knuth's arrow should be used for smaller magnitude numbers, and the chained arrow or hyper operators for larger ones.

Read more about this topic: Knuth's Up-arrow Notation