El_Muntekim   Identity

This is a   complex   Power Tower  type of identity,  giving the   Gelfond's Constant    epi

The storey number (s) is the number of powers in each term of the towers and begins

from two.

eg.   (((x)^y)^x)^y) , implies  that the storey number (s) is 4.

(((((x)^y)^x)^y)^x)^y) ,  implies  that the storey number (s) is 6.

Like the famous Euler Formula  This expression connects the fundamental numbers

(e, pi, i, 1); some integers (2, 3, 4); and Complex Power tower series.

The rules are;

A-   Terms on the numerator alternates as singles;    (x^y^x^y^....) + ( y^x^y^x^.....)

B-   Terms on the denominator alternates as doubles;   (x^x^y^y^x^x^...) + (y^y^x^x^y^y^...)

C-   Swapping  and  y  doesn't make a difference in final result.

D-   (t)   is any integer, such  that;   t ³ 1

E-   All the computations are executed from the innermost to the outermost parenthesis.

F-   Total number of  (x's)  in all the towers = Total number of  (y's)  in all the towers = 2 s

(This provides a  check  for the correctness of the expression)

There are several examples in the following, to make you understand the system better.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

f1=f2 ;  f1 and f2 are identical expressions.

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You can  have  (s)  storeys in general and in each case;  f1 and f2 are  always

identical expressions.

Question: What will happen  as  (s)  approaches to Infinity, while (t) remains a finite Integer..?

Will  the expression  (f1) be  still  identical  to  Gelfond's constant...   ?

= ?