03/07/2008, 06:58 PM

andydude Wrote:I just found the asymptotes of pentation, hexation, heptation, octation, and beyond! And they're fascinating:

...

I suppose you could see this from the integer versions of these operators, but I think the continuous (or if not continuous, mostly real-valued) versions make it easier to see.

First, Andrew, these are really fascinating findings.

andydude Wrote:for all

meaning, in the limit, all hyper-operators return to the successor operation, like the circle of life...

I dare a proof by induction which only needs the integer operations.

Proposition:

If we have a sequence of operations [n] on the natural numbers (>0) that satisfy b[n+1]1=b, b[n+1](x+1)=b[n](b[n+1]x) for n 1, then we can extend the domain of the right operand of [n] to integer k with k 3-n and the only way to do so still satisfying the above conditions and injectivity of the functions f(x)=b[n]x is by b[n](-k)=-k+1 for 0 k n-3.

Proof:

We prove by induction over k that b[n](-k)=-k+1 for all n k+3.

Induction Start k=0:

b[n]1=b=b[n+1]1=b[n](b[n+1]0), for n2,

by injectivity follows

1=b[n+1]0 for n+13=0+3

Induction Step k=k+1:

by induction assumption for nk+3 :

b[n](-k)=-k+1=b[n+1](-k)=b[n](b[n+1]-(k+1))

by injectivity:

-k = b[n+1]-(k+1)

which is the induction assertion:

-(k+1)+1=b[n+1]-(k+1) for n+1k+1+3