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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 (up arrow) is the same as exponentiation:
m up arrow n = m × m × ... × m (n terms) = mn
Two up-arrows together represent a power tower: m up arrowup arrow 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 up arrowup arrow 2 = 2 up arrow 2 = 4
2 up arrowup arrow 3 = 2 up arrow 2 up arrow 2 = 2 up arrow 4 = 16
2 up arrowup arrow 4 = 2 up arrow 2 up arrow 2 up arrow 2 = 2 up arrow 16 = 65536
3 up arrowup arrow 2 = 3 up arrow 3 = 27
3 up arrowup arrow 3 = 3 up arrow 3 up arrow 3 = 3 up arrow 27 = 7625597484987
3 up arrowup arrow 4 = 3 up arrow 3 up arrow 3 up arrow 3 = 3 up arrow 3 up arrow 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 up arrowup arrowup arrow n = m up arrowup arrow m up arrowup arrow...up arrowup arrow m (n terms).
For example:
2 up arrowup arrowup arrow 2 = 2 up arrowup arrow 2 = 4
2 up arrowup arrowup arrow 3 = 2 up arrowup arrow 2 up arrowup arrow 2 = 2 up arrowup arrow 4 = 65 536
2 up arrowup arrowup arrow 4 = 2 up arrowup arrow 2 up arrowup arrow 2 up arrowup arrow 2 = 2 up arrowup arrow 65536 = 2 up arrow 2 up arrow...up arrow 2 (65 536 terms)
3 up arrowup arrowup arrow 2 = 3 up arrowup arrow 3 = 7 625 597 484 987
3 up arrowup arrowup arrow 3 = 3 up arrowup arrow 3 up arrowup arrow 3 = 3 up arrowup arrow 7 625 597 484 987 = 3 up arrow 3 up arrow...up arrow 3 (a power tower 7625597484987 layers high)
3 up arrowup arrowup arrow 4 = 3 up arrowup arrow 3 up arrowup arrow 3 up arrowup arrow 3 = 3 up arrowup arrow 3 up arrowup arrow 7625597484987 = 3 up arrowup arrow 3 up arrow...up arrow 3 (a tower 3 up arrowup arrow 7625597484987 layers high)
Similarly,
m up arrowup arrowup arrowup arrow n = m up arrowup arrowup arrow m up arrowup arrowup arrow...up arrowup arrowup arrow m (n terms)
so that, for example:
2 up arrowup arrowup arrowup arrow 2 = 2 up arrowup arrowup arrow 2 = 4
2 up arrowup arrowup arrowup arrow 3 = 2 up arrowup arrowup arrow 2 up arrowup arrowup arrow 2 = 2 up arrowup arrowup arrow 4 = 2 up arrow 2 up arrow...up arrow 2 (65536 terms)
2 up arrowup arrowup arrowup arrow 4= 2 up arrowup arrowup arrow 2 up arrowup arrowup arrow 2 up arrowup arrowup arrow 2 = 2 up arrowup arrowup arrow 2 up arrow 2 up arrow...up arrow 2 (65536 terms)
3 up arrowup arrowup arrowup arrow 2 = 3 up arrowup arrowup arrow 3 = 3 up arrow 3 up arrow...up arrow 3 (7625597484987 terms)
3 up arrowup arrowup arrowup arrow 3 = 3 up arrowup arrowup arrow 3 up arrowup arrowup arrow 3 = 3 up arrowup arrowup arrow 3 up arrow 3 up arrow...up arrow 3 (7625597484987 terms)
= 3 up arrowup arrow 3 up arrow 3 up arrow...up arrow 3 (3 up arrow 3 up arrow...up arrow 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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