05/31/2011, 09:06 PM

(05/31/2011, 09:05 AM)bo198214 Wrote: Like the Euler number is the argument x where b^x = x for the b where b^x has exactly one fixpoint - in the transition from two fixpoints to no real fixpoint (i.e. ),

the "succeeding Euler number" is the argument x where b[4]x = x for the b where b[4]x has exactly one fixpoint - in the transition from two fixpoints to no real fixpoint (on x>0, because there is always a fixpoint between -2 and 0 for the tetrational.)

This b is around 1.635....

And it looks in the graph as if the maximum of the difference is achieved exactly at this new Euler number.

PS: We also have the analogon to the function , this function has its maximum at x=e and the maximum is e^(1/e). Same with the self-tetra-root it has its maximum at the "succeeding Euler number" 3.08853227... and its value is around 1.635... as Mike found out in this thread.

PS2: Ya and it also has the analogon to the maximum b such that b[4]oo exists (which is e^(1/e)[4]oo = e). So the maximum base for which b[5]oo exists is . Andrew/Nuninho pointed it out in this thread.

All of this was so fascinating, to see such a connection between exponentiation, tetration and pentation. Truly fascinating.