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:
“Read the Bible. Work hard and honestly. And don’t complain.”
—Billy Graham (b. 1918)
“I think, for the rest of my life, I shall refrain from looking up things. It is the most ravenous time-snatcher I know. You pull one book from the shelf, which carries a hint or a reference that sends you posthaste to another book, and that to successive others. It is incredible, the number of books you hopefully open and disappointedly close, only to take down another with the same result.”
—Carolyn Wells (1862–1942)