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Knuth Up-Arrow Notation

Knuth's up-arrow notation is a notation invented by Knuth (1976) to represent large numbers in which evaluation proceeds from the right (Conway and Guy 1996, p. 60):

m^nm·m...m_()_(n)
m^^nm^m^...^m_()_(n)
m^^^nm^^m^^...^^m_()_(n)

For example,

m^n=m^n
(1)
m^^n=m^...^m_()_(n)=m^(m^(·^(·^(·^m))))_()_(n)
(2)
m^^2=m^m_()_(2)=m^m=m^m
(3)
m^^3=m^m^m_()_(3)=m^(m^m)
(4)
=m^m^m=m^(m^m)
(5)
m^^^2=m^^m_()_(2)=m^^m=m^(m^(·^(·^(·^m))))_()_(m)
(6)
m^^^3=m^^m^^m_()_(3)=m^^m^(m^(·^(·^(·^m))))_()_(m)
(7)
=m^...^m_()_(m^(m^(·^(·^(·^m))))_()_(m))=m^(m^(·^(·^(·^m))))_()_(m^(m^(·^(·^(·^m))))_()_(m)).
(8)

m^^n is sometimes called a power tower. The values n^...^_()_(n)n are called Ackermann numbers.

See also

Ackermann Number, Chained Arrow Notation, Down Arrow Notation, Large Number, Power Tower, Ramsey Theory, Steinhaus-Moser Notation

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References

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.

Cite this as:

Weisstein, Eric W. "Knuth Up-Arrow Notation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KnuthUp-ArrowNotation.html

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