Graham's Number
Graham's number itself can not be expressed accurately in Conway chained arrow notation, but by defining the intermediate function, we have: (see functional powers), and
Proof: Applying in order the definition, rule 3, and rule 4, we have:
- (with 64 's)
- (with 64 's)
- (with 64 's)
- (with 65 's)
- (computing as above).
Since f is strictly increasing,
which is the given inequality.
With chain arrows it is very easy to specify a much larger number. For example, note that
which is much greater than Graham's number.
Read more about this topic: Conway Chained Arrow Notation
Famous quotes containing the words graham and/or number:
“The psychologist called him a psychopath. They didnt know what else to call him.”
—Michael Mann, U.S. screenwriter. Michael Mann. Will Graham (William Petersen)
“The more elevated a culture, the richer its language. The number of words and their combinations depends directly on a sum of conceptions and ideas; without the latter there can be no understandings, no definitions, and, as a result, no reason to enrich a language.”
—Anton Pavlovich Chekhov (18601904)