Knuths up-arrow notation

Knuth's up-arrow notation is useful to represent very large numbers with iterated exponentiation. It works in a way similar to standard exponentiation. For example:

<math>2\uparrow\uparrow 3= 2^{(2^{2})}=2^{2^{2}} </math>

Generally:

<math>

\begin{matrix} x\uparrow y &=& x^y &=& x \times x \times x \times\cdots \left(\mbox{y times}\right) \\ x\uparrow\uparrow y &=& x\uparrow x\uparrow x\uparrow\ldots\left(\mbox{y times}\right) &=& x^{x^{x^{\cdots}}} \left(\mbox{y times}\right) \\ x\uparrow\uparrow\uparrow y &=& x\uparrow\uparrow x \uparrow\uparrow x \ldots\left(\mbox{y times}\right)&=&(x^{x^{x^{\cdots}}} \left(\mbox{y times}\right))^{(x^{x^{x^{\cdots}}} \left(\mbox{y times}\right))^{\cdots}} \end{matrix} </math>

Often instead of arrows, in strict ASCII, ^^ is used instead of ↑↑.

See also

• mathworld.wolfram.com/ArrowNotation (http://mathworld.wolfram.com/ArrowNotation)