Knuth's up-arrow notation

In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.

In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc. Various notations have been used to represent hyperoperations. One such notation is ${\displaystyle H_{n}(a,b)}$. Knuth's up-arrow notation ${\displaystyle \uparrow }$ is another. For example:

• the single arrow ${\displaystyle \uparrow }$ represents exponentiation (iterated multiplication) $${\displaystyle 2\uparrow 4=H_{3}(2,4)=2\times (2\times (2\times 2))=2^{4}=16}$$
• the double arrow ${\displaystyle \uparrow \uparrow }$ represents tetration (iterated exponentiation) $${\displaystyle 2\uparrow \uparrow 4=H_{4}(2,4)=2\uparrow (2\uparrow (2\uparrow 2))=2^{2^{2^{2}}}=2^{16}=65,536}$$
• the triple arrow ${\displaystyle \uparrow \uparrow \uparrow }$ represents pentation (iterated tetration) $${\displaystyle {\begin{aligned}2\uparrow \uparrow \uparrow 4&=H_{5}(2,4)\\&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow 4)\\&=\underbrace {2\uparrow (2\uparrow (2\uparrow \cdots ))} \;=\;\underbrace {\;2^{2^{\cdots ^{2}}}} \\&\;\;\;\;\;2\uparrow \uparrow 4{\text{ copies of }}2\;\;\;\;\;{\text{65,536 2s}}\\\end{aligned}}}$$

The general definition of the up-arrow notation is as follows (for ${\displaystyle a\geq 0,n\geq 1,b\geq 0}$): $${\displaystyle a\uparrow ^{n}b=H_{n+2}(a,b)=a[n+2]b.}$$ Here, ${\displaystyle \uparrow ^{n}}$ stands for n arrows, so for example $${\displaystyle 2\uparrow \uparrow \uparrow \uparrow 3=2\uparrow ^{4}3.}$$ The square brackets are another notation for hyperoperations.