- 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
Definition[edit]
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
Basic Properties[edit]
Extension[edit]
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.
Changing Bases[edit]
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: