Knuth’s up arrow notation

Knuth’s up arrow noation is a way of writing numbers which would be unwieldy in standard decimal notation. It expands on the exponentialPlanetmathPlanetmath notation mn=mn. Define m0=1 and mn=m(m[n-1]).

Obviously m1=m1=m, so 32=331=33=27, but 23=222=2221=2(22)=16.

In general, mn=mmm, a tower of height n.

Clearly, this process can be extended: m0=1 and mn=m(m[n-1]).

An alternate notation is to write m(i)n for mi-2 timesn. (i-2 times because then m(2)n=mn and m(1)n=m+n.) Then in general we can define m(i)n=m(i-1)(m(i)(n-1)).

To get a sense of how quickly these numbers grow, 32=33 is more than seven and a half trillion, and the numbers continue to grow much more than exponentially.

Title Knuth’s up arrow notation
Canonical name KnuthsUpArrowNotation
Date of creation 2013-03-22 12:58:43
Last modified on 2013-03-22 12:58:43
Owner Henry (455)
Last modified by Henry (455)
Numerical id 7
Author Henry (455)
Entry type Definition
Classification msc 00A05
Synonym up-arrow
Synonym up arrow
Synonym up-arrow notation
Synonym up arrow notation
Synonym Knuth notation
Related topic ConwaysChainedArrowNotation