User blog comment:Deedlit11/A rigorous definition for pentational arrays/@comment-5150073-20130315231954/@comment-5529393-20130315233447

Well, 3^^^3 & 3 would be expressed as {(X^^^X, 1), (1, 3), (0, 3)} in my notation.

I haven't defined hexational arrays yet, but obviously we can assume that X^^^X has fundamental sequence X^^^p. So the prime block of X^^^X would be:

P_3(X^^^X) = P_3(X^^^3) = P_3(X^^(X^^X)) = P_3(X^^(X^^3)) = P_3(X^^(X^X^X)) = P_3(X^^(X^X^3)) = P_3(X^^(X^(3X^2))) = P_3(X^^(X^(2X^2 + 3X))) = P_3(X^^(X^(2X^2 + 2X + 3))) = P_3(X^^(X^(2X^2 + 2X + 2) 3))

this eventually expands to P_3(X^^f(X)), where f(X) is a 27 term polynomial of the form

2X^(2X^2 + 2X + 2) + 2X^(2X^2 + 2X + 1) + ... + 2X + 3.

You then get P_3((X^^(f(X)-1))^3) and then you start reducing one of the three factors, which will take a LONG time.