- This article is now available on my blog: https://www.sligocki.com/2009/10/07/up-arrow-properties.html
Some useful definitions and properties for Knuth's up-arrow notation
is defined for a, b and n are integers and .
Therefore (with b copies of a, where is right associative) and so it is seen as an extension of the series of operations where is basic exponentiation
We can extend the uparrows to include multiplication and addition as the hyper operator.
This system may be consistently expanded to include multiplication, addition and incrementing:
- (for )
- Proof of consistency by induction.
We will show that Rules 3, 5 and 6 imply rule 4
Assume that for any , then
- by rule 6, rule 3 and assumption
Furthermore, by rule 5
Thus the assumption is true for all
Likewise we can show that Rules 2, 5, 6 imply Rule 3 and that Rules 1, 5, 6 imply Rule 2.
Therefore, Rules 1, 5, 6 imply Rules 4, 5, 6 and so consistently extend the system.
- QED
Clearly some of the properties do not extend.
Todo: How do you change bases.
Example:
- what is n'?
For k = 1:
For k = 2. For all there is a unique such that
- for all sufficiently large n
Examples:
- for all
- for all
Thus the base of a tetration is not very important, they all grow at approximately the same rate eventually.[note 1]
In fact these numbers grow very slowly.
Claim:
Note, the left inequality is easy to prove:
Claim: