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 interesting pattern in hyper-operations Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 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? « Next Oldest | Next Newest »

 Messages In This Thread interesting pattern in hyper-operations - by Base-Acid Tetration - 04/30/2009, 04:15 AM RE: interesting pattern in hyper-operations - by andydude - 04/30/2009, 05:35 AM RE: interesting pattern in hyper-operations - by andydude - 04/30/2009, 05:53 AM RE: interesting pattern in hyper-operations - by BenStandeven - 04/30/2009, 10:41 PM RE: interesting pattern in hyper-operations - by Base-Acid Tetration - 05/01/2009, 10:22 AM RE: interesting pattern in hyper-operations - by bo198214 - 05/01/2009, 01:34 PM RE: interesting pattern in hyper-operations - by BenStandeven - 05/02/2009, 08:11 PM RE: interesting pattern in hyper-operations - by bo198214 - 05/03/2009, 08:20 PM RE: interesting pattern in hyper-operations - by BenStandeven - 05/04/2009, 09:15 PM

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