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:
“Scatter my ashes, strew them in the air:
Lord since thou know’st where all these atoms are,
I’m hopeful thou’lt recover once my dust,
And confident thou’lt raise me with the just.”
—James Graham Marquess of Montrose (1612–1650)
“No Government can be long secure without a formidable Opposition. It reduces their supporters to that tractable number which can be managed by the joint influences of fruition and hope. It offers vengeance to the discontented, and distinction to the ambitious; and employs the energies of aspiring spirits, who otherwise may prove traitors in a division or assassins in a debate.”
—Benjamin Disraeli (1804–1881)