Arrow Notation

A Notation invented by Knuth (1976) to represent Large Numbers in which evaluation proceeds from the right (Conway and Guy 1996, p. 60).

$m\uparrow n$	$\underbrace{m\cdot m\cdots m}_n$
$m\uparrow\uparrow n$	$\underbrace{m\uparrow m\uparrow\cdots\uparrow m}_n$
$m\uparrow\uparrow\uparrow n$	$\underbrace{m\uparrow\uparrow m\uparrow\uparrow\cdots\uparrow\uparrow m}_n$

$\displaystyle m\uparrow n$	$\textstyle =$	$\displaystyle m^n$	(1)
$\displaystyle m\uparrow\uparrow n$	$\textstyle =$	$\displaystyle \underbrace{m\uparrow\cdots\uparrow m}_n=\underbrace{{m}^{{m}^{\cdot^{\cdot^{\cdot^{m}}}}}\!\!}_{n}$
$\displaystyle m\uparrow\uparrow 2$	$\textstyle =$	$\displaystyle \underbrace{m\uparrow m}_2=m\uparrow m=m^m$	(2)
$\displaystyle m\uparrow\uparrow 3$	$\textstyle =$	$\displaystyle \underbrace{m\uparrow m\uparrow m}_3 = m\uparrow(m\uparrow m)$
	$\textstyle =$	$\displaystyle m\uparrow m^m=m^{m^m}$	(3)
$\displaystyle m\uparrow\uparrow\uparrow 2$	$\textstyle =$	$\displaystyle \underbrace{m\uparrow\uparrow m}_2 = m\uparrow\uparrow m = \underbrace{{m}^{{m}^{\cdot^{\cdot^{\cdot^{m}}}}}\!\!}_{m}$	(4)
$\displaystyle m\uparrow\uparrow\uparrow 3$	$\textstyle =$	$\displaystyle \underbrace{m\uparrow\uparrow m\uparrow\uparrow m}_3 = m\uparrow\uparrow\underbrace{{m}^{{m}^{\cdot^{\cdot^{\cdot^{m}}}}}\!\!}_{m}$
	$\textstyle =$	$\displaystyle \underbrace{m\uparrow\cdots\uparrow m}_{\underbrace{{m}^{{m}^{\cd... ...cdot^{m}}}}}\!\!}_{\underbrace{{m}^{{m}^{\cdot^{\cdot^{\cdot^{m}}}}}\!\!}_{m}}.$	(5)

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 59-62, 1996.

Guy, R. K. and Selfridge, J. L. ``The Nesting and Roosting Habits of the Laddered Parenthesis.'' Amer. Math. Monthly 80, 868-876, 1973.

Knuth, D. E. ``Mathematics and Computer Science: Coping with Finiteness. Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations.'' Science 194, 1235-1242, 1976.