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 between addition and multiplication lloyd Junior Fellow  Posts: 10 Threads: 1 Joined: Mar 2011 03/15/2011, 11:21 PM Tommy, the relevance to tetration is that tetration is an attempt to extend the sequence a{1}b, a{2}b, a{3}b... (where these are respectively addition, multiplication, exponentiation) to a{4}b (tetration) -- but another way to extend the sequence is to find the functions where phi (which I use for the value value in curly brackets) is inbetween the integer values we already know. Your guesses that x {0.5} x = x!, and x{y}x = x are wrong and do not follow from the definitions. It's true there is no closed form for these intermediate operations as I defined them yet, and it's true that I didn't know Gauss had invented the arithmetic-geometric mean, if that matters to you. But look at the graphics I posted and tell me you don't think the numerical solutions using these weighted functions look good as interpolated curves. Finding closed forms will be very difficult, as already the closed form for Gauss's mean involves some tough steps. (03/15/2011, 10:22 PM)tommy1729 Wrote: if i understand this confusing thread well x {0.5} x = x ! in fact x {y} x = x at least following from ( quote ) In other words, for a {0.25} b, with a