05/01/2009, 01:34 PM
Tetratophile Wrote: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.
I proved your conjecture in the thread that Andrew already referred to. Its the post:
http://math.eretrandre.org/tetrationforu...94#pid1494
BenStandeven Wrote:It should be possible to choose the "natural" versions of the higher operations so that they are approximately linear on the intervals in question, since (e ^k^ x) + 1 = e ^k-1^ (e ^k^ x) = e ^k^ (x+1)
We use notation e [k] x here on the forum for the k-th hyperoperation.
Quote:would be approximately true by induction, and we always have e ^k^ 0 = 1, and e ^^ x = x+1 approximately on [-1, 0]. The degree of approximation might decay a bit at each level, I suppose.
Hm, you mean there is a function that is closest (say by maximum difference or by area between graphs) to the linear function on the interval [-1,0] among the solutions of f(x+1)=e[k]f(x)?
