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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).

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

For example,

$\displaystyle m\uparrow n$ $\textstyle =$ $\displaystyle m^n$ (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)

$m\uparrow\uparrow n$ is sometimes called a Power Tower. The values \(n\underbrace{\uparrow\cdots\uparrow}_n n\) are called Ackermann Numbers.

See also Ackermann Number, Chained Arrow Notation, Down Arrow Notation, Large Number, Power Tower, Steinhaus-Moser Notation


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.

Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 11 and 226-229, 1991.

© 1996-9 Eric W. Weisstein