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    Knuth's up-arrow notation

    A notation for large numbers developed by the American mathematician Donald Knuth (1938-) in 1976. A single up-arrow () is the same as exponentiation:
    m n = m × m × ... × m (n terms) = mn
    Two up-arrows together represent a power tower: m n = m m^m^...^m (a tower of height n), which is the same as the operation known as hyper4 or tetration. This can very rapidly generate huge numbers. For example:
    2 2 = 2 2 = 4
    2 3 = 2 2 2 = 2 4 = 16
    2 4 = 2 2 2 2 = 2 16 = 65536
    3 2 = 3 3 = 27
    3 3 = 3 3 3 = 3 27 = 7625597484987
    3 4 = 3 3 3 3 = 3 3 27 = 37625597484987
    Three up-arrows together represent a still more vastly powerful operator, equivalent to hyper5 or pentation, or a power tower of power towers:
    m n = m m ... m (n terms).
    For example:
    2 2 = 2 2 = 4
    2 3 = 2 2 2 = 2 4 = 65 536
    2 4 = 2 2 2 2 = 2 65536 = 2 2 ... 2 (65 536 terms)
    3 2 = 3 3 = 7 625 597 484 987
    3 3 = 3 3 3 = 3 7 625 597 484 987 = 3 3 ... 3 (a power tower 7625597484987 layers high)
    3 4 = 3 3 3 3 = 3 3 7625597484987 = 3 3 ... 3 (a tower 3 7625597484987 layers high)
    Similarly,
    m n = m m ... m (n terms)
    so that, for example:
    2 2 = 2 2 = 4
    2 3 = 2 2 2 = 2 4 = 2 2 ... 2 (65536 terms)
    2 4= 2 2 2 2 = 2 2 2 ... 2 (65536 terms)
    3 2 = 3 3 = 3 3 ... 3 (7625597484987 terms)
    3 3 = 3 3 3 = 3 3 3 ... 3 (7625597484987 terms)
    = 3 3 3 ... 3 (3 3 ... 3 (7625597484987 terms) terms)
    Even up-arrow notation becomes cumbersome, however, when faced with staggeringly large numbers such as Graham's number. For such cases, more extensible systems such as Conway's chained arrow notation or Steinhaus-Moser notation are better suited. See also the Ackermann function, to which up-arrow notation is closely related.


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       • MATHEMATICS



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