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El_Muntekim Identity

This is a complex Power Tower type of identity, giving the Gelfond's Constant e^pi.

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 x 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.