04/30/2009, 04:15 AM
(This post was last modified: 04/30/2009, 04:34 AM by Base-Acid Tetration.)

I just wanted to know if pentation would have any asymptotes like tetration does and I just discovered this interesting property.

So, tet(1) for base b = b, tet(0) = 1 and tet(-1) = 0, so tetra-logarithm (superlogarithm) of b is 1, tetlog(1) is 0, and tetlog(0)=-1.

pent(1) = b for any b, pent(0) = tetlog(pent(1)) = tetlog(b) = 1,

pent(-1) = tetlog(1) = 0,

pent(-2) = -1.

pentlog(b) = 1

pentlog(1) = 0

pentlog(0) = -1

pentlog(-1) = -2

Continuing this for higher operations (verification is left for the reader), for base greater than 1,

for hexation, hex(1) = b, hex(0)=1, hex(-1)=0, hex(-2)=(-1), hex(-3)=-2;

for heptation, hept(-1) = 0, hept(-2)=-1, hept(-3)=-2, hept(-4)=-3;

oct(-5)=-4, non(-6)=-5, dec(-7)=-6, ...

So here is my conjecture (theorem?)... for n>=3, b[n]-n+3 = -n+4; for n>=4, b[n]-n+2=-n+3, etc.

If you graph these hyper operation functions for integers, you will notice a linear-ish part in a larger domain for increasing n. (specifically around the domain [-n+3,0]), so my conjecture can be stated as:

for k>=3, we have b[k]n=n+1 for any natural n which is in the interval [-k+3,0].

What is the implication of the growing quasi-linear part for the real or complex analytic extensions of those higher hyper-operations pentation, hexation, etc? Is it a good thing or a bad thing?

Also would pentation have any asymptotes?

So, tet(1) for base b = b, tet(0) = 1 and tet(-1) = 0, so tetra-logarithm (superlogarithm) of b is 1, tetlog(1) is 0, and tetlog(0)=-1.

pent(1) = b for any b, pent(0) = tetlog(pent(1)) = tetlog(b) = 1,

pent(-1) = tetlog(1) = 0,

pent(-2) = -1.

pentlog(b) = 1

pentlog(1) = 0

pentlog(0) = -1

pentlog(-1) = -2

Continuing this for higher operations (verification is left for the reader), for base greater than 1,

for hexation, hex(1) = b, hex(0)=1, hex(-1)=0, hex(-2)=(-1), hex(-3)=-2;

for heptation, hept(-1) = 0, hept(-2)=-1, hept(-3)=-2, hept(-4)=-3;

oct(-5)=-4, non(-6)=-5, dec(-7)=-6, ...

So here is my conjecture (theorem?)... for n>=3, b[n]-n+3 = -n+4; for n>=4, b[n]-n+2=-n+3, etc.

If you graph these hyper operation functions for integers, you will notice a linear-ish part in a larger domain for increasing n. (specifically around the domain [-n+3,0]), so my conjecture can be stated as:

for k>=3, we have b[k]n=n+1 for any natural n which is in the interval [-k+3,0].

What is the implication of the growing quasi-linear part for the real or complex analytic extensions of those higher hyper-operations pentation, hexation, etc? Is it a good thing or a bad thing?

Also would pentation have any asymptotes?